Use the above summary and do not change anything without reading the according section.
Jungle route is bluebuff wolves ghosts redbuff golems.
Use your initial Health Potion when heading for a gank.
Save enough mana for your ganks.
You may donate second and should donate third blue buff to your mid lane.

This guide is designed to get Trinity Force as fast as possible and continue with any ad or hybrid build. I basically want to explain the laning phase / jungling here, so at some points I gonna state things to affect late game mainly and do not fix much there.

To get Trinity Force early we need gold. Poppy is a bad farmer in lane so unless you get kills you will take hours to afford it. Thats why I suggest jungling, where you get Berserker's Greaves and Trinity Force between 15 and 25 minutes into the game dependent on how successful you gank.

Pros:

Anti carry
Strong ganks on side lanes.
Strong tower dives.
Strong Baron/Dragon steal.
Strong chase.

Cons:

Weak ganks on mid lane.
Slightly vulnerable to counter jungling.
Vulnerable to dots.

Glyphs

Greater Glyph of Scaling Magic Resist, Greater Glyph of Cooldown Reduction, Greater Glyph of Scaling Cooldown Reduction, Greater Glyph of Scaling Ability Power or Greater Glyph of Ability Power: Not relevant for the jungling.

Marks

Greater Mark of Attack Speed: they give the most speed up in the jungle as they multiply with Paragon of Demacias AD. Those are essential.

Quintessences

Greater Quintessence of Life Steal: a great way to sustain in jungle, that doesn't loose its value later on. They allow you to start the game with a Dagger.
Greater Quintessence of Attack Speed: an alternative but you have to start with Vampiric Scepter then. Results in slower jungle and you can't afford level 2 boots directly when failing your first gank. But still it's possible.

Seals

Greater Seal of Regeneration, Greater Seal of Armor or Greater Seal of Vigor all keep your life up in jungle.
Greater Seal of Armor is standard and you already got them probably. Also they are strongest in later game teamfights, but weaker for the early jungle purposes.
Greater Seal of Vigor / Greater Seal of Regeneration give you more sustain. Break even is at level 4 (!!). You hit level 2 directly when finishing the blue buff camp and till then there wasn't much time to regenerate life, so its basically level 2 and 3 where Vigor is stronger than Regeneration.
Keeping that in mind I experimented to find the proper balance between those runes and ended up using the numbers above. However it is also a matter of personal preference so feel free to adjust, keeping the above in mind.

If you feel you need more mana regeneration add some Greater Seal of Scaling Mana Regeneration, and switch to some more Greater Seal of Regeneration to keep your early sustain up, but it is better if you learn to handle your mana without.

Paragon of Demacia helps best in jungle as it provides both armor and damage, and the damage it provides allows for life steal and doesn't need mana. For ganking it allows you to hit a wall with your Heroic Charge more easily due to the increased run speed. Although Devastating Blow provides more damage during ganks hitting the wall with Heroic Charge is more important as it allows your mates to catch up and deal damage as well. Nearly every gank where you hit the wall with Heroic Charge successes while nearly every gank where you NOT hit the wall with Heroic Charge fails.

At level 3 you want to have all 3 abilities, the order in which you skill them is not that important.

Starting with Heroic Charge is optimum for killing blue buff (and also for lvl 1 teamfights). If perfectly executed a quite standard pull (with ability + 1 autoattack) lets you kill blue buff with taking only 1 or even no hit from it:
When the Golem is pulled hit it once, run back a bit hit it again and then directly charge it into the wall where it spawned. It is quite difficult because you want to use your Heroic Charge at maximum range and you have to hit the direction, but if you manage to do it that way it gets stunned and has to restart its walk to the point where your pulling midlaner looses aggro. Then try and stay in front of the golem so it has to walk around you for some extra time. That gives you (almost) enough time to kill it. If not perfectly executed you will take some more damage, but thats not too bad.
Any other of your skills allows the golem to deal more damage to you and is not significantly faster.
In case you intend to pick the second blue buff for your self you may pick level 2 Devastating Blow at level 5 or 7 for slightly stronger ganks.

The mastery page shown above does not require much explanation i think. It is designed to boost early AD/AS and jungle speed/sustain. That is:
Sustain: Vampirism , Hardiness , Vigor
Jungle: Bucher, Tough Skin , Bladed Armor
early AD/AS: Brute Force , Alacrity
summoner spells: Summoner's Resolve for Smite, Summoner's Wraith if you pick one of the affected spells
just strong and on the way: Executioner , Weapon Expertise
rest: doesn't matter that much, I use those shown above.

Trinity Force is by far the best item on Poppy. Any stat it gives, including the mana, helps Poppy and the unique heavily boosts your damage. It's just incredible.
Because it is that incredible I want to have it as fast as possible, and that is the main objective of my build. So I head for it directly after boots.

Starting Item

Dagger: I've experimented a lot and found out that this combined with the runes/masteries above allows for the fastest jungle with enough sustain. Also it build into Berserker's Greaves, allowing level 2 boots even if you fail your first gank (500 gold). If you do choose other boots it builds into Zeal -> Trinity Force.

Vampiric Scepter: The second best combination is Vampiric Scepter and Greater Quintessence of Attack Speed, but it is a bit slower jungle for a slight overkill in sustain. Also it does not build into boots nor Trinity Force, so it forces you to earn 450 more gold until you get your Trinity Force.

Doran's Blade: Possible but the others are stronger.

Boots

Berserker's Greaves: I recommend those. As I skill Paragon of Demacia first the additional attack speed greatly increases my early damage and life steal. They are also the cheepest and they build out of our starting item.

Ionian Boots of Ludicity: Late game some cooldown reduction is great, but it is not necessary in early game and so you better get it from some other item.

Sorcerers Boots: They also do not help much early on, later they would be fine.

Mercury Threads: Tenacity is cool on Poppy, as she is vulnerable to cc. However those are the most expensive boots and they do not increase our jungle speed and only helps a bit when ganking. So I suggest to get Tenacity from Cloak and Dagger or Moonflair Spellblade later on.

Trinity Force

Zeal: I pick this one first as the attack speed multiplies with the AD from Paragon of Demacia which is my highest priority skill, and by that it increases my jungle speed and gank potential most. Also the run speed helps placing Heroic Charge through ganks and allows to keep up for further damage.

Phage: Within the jungle this one increases my damage more than Sheen, because i rarely use abilities there due to mana issues. For ganks Sheen would provide more damage, but the Phage slow allows you and your mates to keep close enough for further damage which is more useful in most cases.

Sheen: the last part of Trinity Force, and as Trinity Force only costs 300 gold to combine I rarely build this one explicitly but most of the times just finish Trinity Force directly.
However if you prefer to build Sheen before Phage do so, it is also viable, especially when using Greater Glyph of Ability Power or Greater Glyph of Scaling Ability Power.

finish Trinity Force!!!!

the rest

basically here my build ends, however the game by far doesn't. You now may continue with any other build you want to. As most builds do contain Trinity Force just choose your preferred guide (other than this) to find out what to build next. AD, hybrid or bruicer builds do fit better because your runes do support them. I give some suggestions how those builds could possibly look like, but I do not explain why, I leave that to other guides.

AD: Maw of Malmortius Bloodthirster Infinity Edge Phantom Dancer ( Cloak and Dagger)

Hybrid: Hextech Gunblade Guinsoo's Rageblade Nashor's Tooth Malady

Offtank: Warmog's Armor Atma's Impaler Force of Nature Zeke's Herald

Pick:
Smite is necessary for Poppy jungle. The other one is your choice, don't forget to pick the according mastery (except for Flash):

Choose:
Flash: after Diplomatic Immunity tower dive you need to get out of tower range fast, so Flash is great, also it allows you to position yourself for Heroic Charge or to escape. You can't pick the according mastery.
Heal: due to Valiant Fighter Poppy takes half damage when near dead. With heal you remain in that state even longer, so heal on Poppy has near double the effect it has on other champs.
Ghost: You are a melee, this one is great for chasing.
Exhaust: Diplomatic Immunity + Exhaust = easy kill on a non tank of your choice. Also good for early ganks.
Surge: Especially in the hybrid build (but also in the other builds) you profit both from ap and as. Additionally having burst when under Diplomatic Immunity protection is great.
Cleanse: being crowd controlled is never nice, but I prefer Cloak and Dagger, Banshee's Vail or Quicksilver Stash.
Ignite: Your Diplomatic Immunity increases its damage and it is nice to pick up kills you would miss otherwise. If you tower dive this might let you leave turret range while still under Diplomatic Immunity protection while letting the dot kill your target.

its a question of personal preference.

Don't pick:
Revive: it is weak, only possibly valuable on some certain champs that either defend a turret against multiple remaining opponents or have a way to port back into the fight ( Pantheon Twisted Fate)
Teleport: jungle waits for you, so you do not have to hurry to get back. that is something for solo laners who cant let there lane alone for too long.
Promote: you need to meet a Siege minion to use it, they are not in the jungle and when ganking you need some luck to have one present. And even if you meet one it is not great advantage to use Promote so why pick it?
Clairvoyance: it allows you to charge someone who tries to hide in the brush (many brushes are at walls), but ... let a support pick it, all of the suggested spells above help you more.
Clarity: if you take care of your mana you shouldn't get too much trouble with it.

First clear

As stated above take the standard jungle route:
Bluebuff -> Wolves -> Ghosts -> Redbuff -> Double Golems -> Health Potion Gank ->recall or clear once more.
Use Paragon of Demacia (active) when approaching a camp (while under the effect of Blue Buff) to both use the run speed and initiate with max stacks. For most camps that is just after killing the last monster of the previous camp. For ghosts it is approximately when having crossed the lane.
Ganking earlier than the proposed is possible as you mainly need Heroic Charge and you pick it at level 1. Do so if there is a good opportunity, but do not enforce it.

the rest of laning phase

Now there is no fixed path to run along anymore, just some hints to keep in mind.

Ganking mid is nasty, as there is only a bit of opposite side wall to smash opponents in. It's hard to hit. To effectively help your mid laner you want to donate as many blue buffs to him as possible.

You do not have blue buff any more so you have to watch your mana. You need mana for ganks so only use mana for jungle farm if you have enough (near full) or if you are sure you won't go for a gank before the next recall. Otherwise just rely on your auto attacks, they will do enough damage now thanks to Paragon of Demacia and Berserker's Greaves (and possibly Zeal+ Phage).

At very low hp your passive nearly halfs even the jungle creeps damage. So your life steal outheals the jungle creeps damage which allows you to still may farm a bit (if you are sure not to be attacked by opponents), but do not expect to heal yourself up by that until you got your Vampiric Scepter.

Counter Jungling

It heavily depends on the opponents jungler (champ, player + build) whether you can 1v1 them or not, so try to get a feeling for your strength before hunting opponents junglers. Stealing opponents jungle farm / buffs is easy. If you invest some mana you get those monsters down fast.
However for any counter jungling you have to use mana you can't use for ganking lanes then. So just farming your own jungle a bit more is a viable alternative.
Keep in mind: your main goal is to feed yourself while helping the lanes if neccessary. It is not your job to shut down any opponent, (if fed you can easily do that late game with Diplomatic Immunity).

Warding

You do not use Wriggle's Lantern so you do not have a build in ward engine. Additionally Poppy is carry-like so gold on you is worth more than gold on many other champs so it would be better if your support did all the warding. However he can't and maybe he does not even care, and in that case warding yourself is better than no warding. Early game your mates should ward their lanes. You should only need to ward opponents blue and red buff, if you want to counterjungle. Late game your support should be able to reach most important warding spots, but it is quite much to do all alone so helping a bit doesn't hurt.