Makes you invincible against a lane without Ignite or a Healing Debuff, abuse with your passive to never have to leave lane and outfarm/exp your lane opponent.

Skilling Order
Masteries + Runes

Greater Mark of Insight
Magic Penetration, especially flat, is strong early game.

Greater Seal of Resilience
Durability in lane early game and a bit of armor late.

Greater Glyph of Warding
Same as Armor, you'll need this in lane.

Greater Quintessence of Swiftness
Movement Speed is strong, especially as utility, allowing you to trade with your opponent and get out with minimal retaliation.

9/0/21
Summoner Abilities

Flash, it's overpowered, it can be used offensively or defensively, don't waste it though, only use it when it's absolutely needed.

Ignite, Torment increases Ignite, making it core to Swain's gameplay.
Items

You'll want to start with a Faerie Charm, Health Potionx4 and Mana Potionx2. This will lead into a Philosopher's Stone and Boots of Speed on your first back. Then rush a Catalyst the Protector and Sorcerer's Shoes.

Turn your Catalyst into a Rod of Ages if you can by 20-21 minutes. If not, save it for a Banshee's Veil, then rush a Rabadon's Deathcap, followed by a Zhonya's Hourglass

Now you maybe like, why Hourglass? Well Billy, your ult persists through Hourglass, so you can hit it an emergency and still do damage, and get the last ticks of healing (It doesn't proc the healing while in it.)

Past this, it's situational, you can either get Will of the Ancients, Rylai's Crystal Scepter, or Spirit Visage.

However, be sure to upgrade your Philo into an Eleisa's Miracle for the Tenacity.
What not to get

Chalice of Harmony, it's a noob trap. It only is effective when you have preexisting Mp5, which you wouldn't have enough to make it greater than a Catalyst.

Archangel's Staff, a good item. Don't get me wrong, AA is a really good item, for those that can afford to go glass cannon. You won't need the mana or the regen if you play smart, and RoA gives you double the stats and health on top of it.

Along with those two, use common sense, if you're getting blown up by Melee, get Hourglass a bit sooner, if you're getting owned by BEAR TIBBERS, get a BVeil.
Farming

Never push the lane unless you're confident that you're not going to get ganked. IE by a jungler, or sidelane.

Last hit with Swain, make sure you always get the passive proc, only farm with your Ulti + W if you need the health and you're allowed to free farm.
Working in the team

Follow the tank, always, never be alone past the 30 minute mark, even less depending if lane phase ended. Never initiate, you can't be caught, once the fight gets committed to, use Nevermove, use your EQ combo and flash in with Ravenous Flock, using Ignite on the target of your combo.
Jungling

You can't jungle, however, once you get lv6, you should be farming Wraith camp with your ulti + W and grab Blue buff, it's insanely strong for Swain, as it allows to maintain his R.
Pros / Cons

Pros
-Insane lane staying
-Durable
-High damage over time
-Is a huge threat in team fights

Cons
-Shut down by ignite
-Susceptible to high burst
-Nevermove is hard to land if inexperienced
Unique Skills

Once you hit 6 you never have to leave lane other to gank, you barely ever have to blue pill to base.

You have a single target Vlad ultimate.

Swain is one of the funnest AP carries out there, and he's durable, use that to your advantage.