Maybe I'm not the best person to be making a guide, but I sincerely hope that it will be useful for some of you. At the beginning of season 7 I was placed in bronze 5 and since then I have climbed to Silver 4. The purpose of this build is to finish the game in the 25-35 minute mark. You will have high amounts of damage and sustain that will ensure you can take down both squishy and tanky enemies. I'm putting a lot of work and experience into this guide, so I hope that you will read the whole thing!
PS Sorry for not including too many images in the guide, it seemed very time-consuming.

At level one, depending on the type of target you're facing, take either Q or E. Q against less mobile champions that will have a hard time getting out of it's range and will take plenty of damage. E is against targets who have more mobility, primarily melee champions. If you take E first you can take your enemy down to half health with poking by level 2. Don't expect to get a kills until level three tho. If you're facing off against a really squishy target, you might want to go for a Q -> W combo if you think you can kill them at level 2 with ignite. Farming is slightly hard in the first few levels unless you take attack damage marks, but that shouldn't worry you. Once you hit level 6 and get your first item (Hextech GLP-800) the game becomes a breeze.

Farming is absolutely essential to Swain. You can give up plenty of kills and be total dunce, but as long as you keep farming and sustain your mana, you will not fall behind. Your farming should include W + Q against waves without counter minions and W + Q + R for waves with counter minions. In the first few levels you should not push in the wave as you have low mobility and a high chance of getting ganked. After you reach level 6 your primary goal should be taking your enemy's turret. You might pick up a few kills here and there, but it isn't kills that win the game - it's objectives.

If you spot an open lane with enemy minions building up, not only is it a good source of gold, but you are likely to prevent your enemy from pushing and taking towers. It's important to have presence around the map, but you NEED to keep your own lane clear. Having your lane won and no longer contested could both save the game and win it.

Ever since the rune update (today - 8th of November 2017) there hasn't really been a very compatible keystone rune, not that deathfire was all that amazing. So this is where Kelptomancy comes in. All you have to do is use a spell then autoattack your foe. Since you'll most likely be facing a melee champion this shouldn't be much of a problem.
My rune build is centered around the long laning phase and gaining a very large gold advantage. The greater and lesser runes will save you 950 gold that you would probably eventually spend. Also the extra movement speed on your boots and the early stopwatch can come be quite handy in case of a gank (you really shouldn't let them gank you!!)
The additional gold from triumph is nice, but nothing too great - what you thrive on here is the 15% missing health regenerated - it makes all the difference!
For the lesser rune you can choose whatever fits your gameplay best.

According to most guides RoA is the standard first item for Swain, but I beg to differ. The recommended item by riot, Hextech GLP-800 is absolutely amazing, and a bit underrated. The early damage it provides is simply amazing, not to mention the fantastic synergy it has with Lyandry's Torment. The magic penetration increases the damage dealt by GLP-800's active and that same active procs Lyandry's improved passive. If you think that the active doesn't justify the purchase over RoA, think of it this way - you get an extra damage ability in your kit.

Sorcerer's shoes are a must unless the enemy team has no magic resistance at all and has an AD jungler such as Xin Zhao or Master Yi. In which case Ninja Tabi will prove extremely useful.

Lyandry's torment isn't great only because of it's passive, but also because of the 15 magic penetration you gain. It is essential to your build and synergises well with every aspect of it.

For my fourth item I would usually buy Rabadon's Deathcap if I am ahead. It boosts your damage and it boosts your healing considerably. Again, it has great synergy with every aspect of your build, however if you're falling behind a Zhonya's might be better. It really depends on your needs in the game. If the enemy team has 3 tanks (Top, Jungle and Support) then you might want to pick up a Void Staff.

The game should be coming to a close by your 4th item, however any of the aforementioned items will fit nicely into the build. Luden's Echo is good if you still need damage, which I really doubt. If your ADC is incompetent and not pushing at all, then you should look into getting a Nashor's Tooth.

Spirit Visage can come in as a 4th or 5th item against an AP heavy team, but the bonuses Rabadon's Deathcap provides in a typical game are greater.

If you have a good team comp and some left over gold get The Dark Seal. It's going to help you snowball even more.

Always look to push up lanes. While you are an important asset in your team composition, you could be pushing out a lane. It comes down to the situation. If your entire team is simply stalling and not engaging, then you should probably go to a pushed out lane and try to take objectives. If however, your team is engaging you can also either join them or defend your towers from the oncoming siege. Always try to focus down the ADC when defending your tower. You will have enough damage to kill them.

Teleport is good against champions which you won't able to kill such as Dr. Mundo and Tryndamere. They also have very good wave clear and pushing so, the only way to match them is to have teleport. You should also consider taking this if your ADC is Jhin who has little pushing capabilities. You will find that you have little use for teleport as long as you shove the wave before backing.

Ignite will come in handy when dealing with squishy melee champions. When facing off against a low threat champion such as Darius, Garen or Yorick it can help you snowball extremely well. You should be picking this spell against melee champions with low mobility.

Ghost serves a similar function as both ignite and flash. The beauty of this summoner spell is that you can use it both offensively and defensively. Someone who is more familiar with Swain and knows how to play safely will probably find ghost the most useful summoner spell. It allows you to chase down targets with little health, kite around team fights and survive ganks.

Flash Though flash seems like a necessity to most people - it isn't. It's just a summoner spell like all the rest and as such has limited utility. You can easily replace flash with ghost, but if you haven't gotten used to playing Swain and avoiding ganks, then flash will keep you safer.

Exhaust is particularly viable against fighters who tend to mow you down or have a lot of mobility (Fiora, Talon, Volibear). It's a very situational pick.

Pros

Amazing Sustain
Amazing Damage
Great Teamfights
Amazing Farming
Decent Crowd Control

Cons

Decent Sieging
No pushing capabilities other than farming