Devastating blow is essential for harassing enemy players right away, getting one off early game deals 150-200 damage and puts a real fear into your enemies. (Doran's ring is not commonly used with poppy to start but i have had good success with it, keeps you in the lane longer without having to go back for manna)
Once level 2 you now have the ability to pull off early ganks with your bottom teammate. Communication is necessary with Poppy because you will need a wall-slam with heroic charge for the bonus damage and stun to successfully eliminate your squishiest opponent very early in game. If your partner has any slows, snares or stuns to use immediately after you guys will be well on your way to dominating bottom. Just remember Poppy is slow early game and to play safe and don't dive in. At level 3 you'll have Paragon, which passive stacks get you that little bit more oomf to put some scare into your enemies. I would not recommend activating it low level as it is manna costly and Poppy will need items to boost her manna before using all her abilities.

By level 4 or 5 if played correctly with your bottom team mate you should have at least one kill by now and hopefully have your first item "sheen" Now with some manna behind you you can use all 3 abilities to give a massive hit with your Devastating Blow. My preferred combo is W E Q, Active paragon will give you an attack bonus and speed boost to run up to your enemies closing the gap on ranged, then with your heroic charge you catch them by surprise with a fast dash followed by devastating blow to deal a huge hit which will almost always send them running for their tower. Devastating blow counts as a basic attack and stacks with sheen's passive to give you bonus damage on a basic after using an ability.

Level 6. Now with your ultimate you can go through the jungle to pull ganks on other lanes to help your team, picking out the weakest of the enemies to deal bonus damage to them and ignoring damage from all others, with a well timed wall slam with heroic charge on an Ult locked enemy you can pull off a 2-3 hit kill.

Now hopefully by around level 8-10 you will have Trinity Force, and this is where Poppy starts to level extremely slowly. Getting TIAMAT will allow you to farm minions and keep your leveling pace up, going for kills only when necessary, staying back and trying your hardest not to die. These levels here are crucial to your success and if you start getting cocky you will get yourself killed and fall behind. Remember to pick out the weakest (Ashe and similar champions can be 2 hit no problem) but get out right after, Poppy is usually focused in team battles because people are scared of her huge damage output and want to be rid of her ASAP.
At 11 you will get your second level Ult and be able to deal even more damage to your targeted enemy, this is where you start moving with your team.
You can initiate team battles with your ult active, but now targeting the weakest may not be a good choice, if the enemy dies you are no longer immune to other damage.
Now you are able to pick on tanks, getting your team a huge lead on taking their tanks hp down and good for tanking herself with her ult active. watch your ultimate's timing though do not let it run out while you are being focused or certain death awaits. Target, Combo, grab another basic attack or two then get the hell out of there because they will all be hot to kill you. Now with people chasing you, you become a perfect lure for your team to pick off chasing enemies.

If you read the whole last chapter, follow those engaging instructions and you should be fine to get kills and stay alive throughout these levels. Follow the item buy order too "Maw of Malmortius" is an optional item, if you are getting picked on by magic characters the magic resist will help immensely, and the extra attack damage is always welcome to Poppy. I would not recommend AP because her Devastating Blow has a bonus damage cap and the bonus AP damage will get restricted by that cap, however your base attack damage will not cap out any restrictions on the skill.
Now With the Ravenous Hydra and hopefully the Sanguine Blade, your life steal will keep you pushing through fights, especially if minions are around, Hydras AOE strikes will heal you insanely to keep you in a fight for a long time.