I'll make this quick, short and sweet...
You can carry even a pretty horrible team with this guide, as long as you have one other decent person. You will be almost unkillable even when focused.

Starting items:
Boots and hpots are pretty standard, good for mobility and sustain early on. They are important because Swain starts out squishy and you will probably be taking a little damage while harassing. However you might also want to start with a Doran's Ring, depending on who you're midding against. Champions who aren't going harass you as hard, go with the ring.
Sapphire Crystal: I've tried this and it's just not as good because as long as you're lasting hitting at a somewhat decent rate carrion renewal will take care of you. Your CD's are too high early game anyway, so you don't need that extra mana until later.
Meki Pendant: No, don't build ToG on Swain, you aren't getting archangel's staff. Better choices.
Lanes: Either solo top or mid. Your best bet is against a squishy so you can get first blood, but you won't be killing anyone until lvl 4 anyway.
Early game: Lvl 1-3 focus on last hitting, harassing with e. Once you've reached level 4, get your q. If he is 3/4 health, try to land a nevermove. If you land it, follow by an E+Q Ignite with autoattacks and you should be able to finish him. If you don't think you can land the kill, harass with E+Q and continue to last hit.
Swain's DoT is usually underestimated. You will play best by landing your nevermoves. If they're standing still, shoot it behind them where they will run. The range is very good so take advantage of that. It's almost always better to overshoot nevermove than to undershoot it.
The Master Tactician-Play off of your opponents mistakes. If they run at you q them and run away. When they start to run back (they will), land an easy nevermove. When you're beating down the enemy tower don't be afraid to dive a little if there are enemy creeps under it because they will heal you as long as you're in Ravenous Flock. Shoot the nevermove and if it misses then don't fight. Play aggressive but be cogniscent. If your ubers not up, you're going to die really fast.
Mid Game: Once you have your catalyst, and your uber, you have a ridiculous amount of sustain. You will be able to out lane people like Brand or Anivia, who previously could wqe you in one shot. With a couple Doran's Rings and your catalyst, your only problem should be managing mana.
Late game: In team fights, you might find yourself running around a lot like singed. If you start to get focused, uber away q then w. Start dropping bombs. Most of the time you should be bursting down their carry though, e+q then w in the way they are moving. Always remember ignite procs with Torment.
Oh yeah, I almost forgot to mention. . . NO ONE NEEDS BLUE BUFF MORE THAN SWAIN. Not Nidalee, not Ryze, absolutely no one. Swain's uber drains mana so fast in team fights, and if you don't have uber on you risk getting focused down and killed. However, if you're doing good and feel confident, let someone else get it who is not doing as well. You will do well no matter what, blue buff just keeps you in the fight longer.
Kiting is extremely important while using Swain. You will feel like you're playing Singed almost. If you happen to overextend and get ganked, Ravenous Flock followed by a Decrepify on their melee. Immediately after drop a nevermove directly in front of the group. Yeah, I know I said Swain has no escape but he does have amazing escapability as long as you equip my knowledge. Just don't give up, you can escape a lot of ganks because of your CC, uber, and tankiness.

Pretty self-explanatory, you're a caster so you want mpen and ap. You could also branch into utility more than offense if you want to play more passively.
Note: I don't like the mastery for ignite, it's just not as good. I'm also bitter because the old mastery gave 10 ap instead of 5 ap and 5 ad.

I've switched this up quite a bit in the past but ultimately found this to be your best bet. You could start with Q as it's your best farming tool or W because it has AoE creep damage. Try not to farm with your W though.
Maxing q first gives you the most reliable damage output. Yes they can break the tether, but you probably won't land every nevermove.

Your best bet is easily Ignite because it procs with Torment. Flash is good because Swain has no escape.
Cleanse- The only other alternative in my eyes. It gets rid of Ignite now, which shuts down Swain's healing hard. I still prefer ignite, it helps you burst down a single target (which is what Swain excels at).

Swain is a pretty terrible farmer. His last hit animation is almost as bad as Karthus', but he doesn't really have an ability to make up for it. Earlier on, q is your best farming tool. Later on when you have your uber, Ravenous flock and then drop a nevermove to finish off the creeps.

This is a big one for me, I've done almost every build on Swain you can think of, besides AD swain. That being said, don't always follow the build order I have posted. Swain really does NOT have any core items. But IMHO, building tanky/AP is what compliments our master tactician the most. I have my items set up as 2 tanky AP items followed by a defensive item. Either Frozen Heart or Banshee's Veil depending on the enemy team. However if you're doing extremely well then get a Deathcap. More AP=more heal.
With this build, you are not seen as the AP carry because your AP won't be that high. But you will be racking up kills like crazy, I promise.
Deathfire Grasp- I used to hate this item because it has an active. I've started using it more and more because it procs with Swain's E. What an amazing item to have on Swain, especially on the Twisted Treeline. After getting it a couple of games in a row, the active became easier and easier to use. I highly recommend this for the Warmog's/Atma bruisers, hard counter. Plus it gives you mana regen and CDR which works out great. 60 AP isn't a lot but you're not Veigar, are you? You're the ******** Master Tactician and you can initiate a fight as well any other Tank and annihilate twice as much.
RoA- What I believe is the best compliment to Swain. If Swain has a core item, it's this. Always get this item unless you're team literally can't do any damage.
Rylai's- It makes all your abilities have CC, and makes your uber slow 3 at once. Not to mention it increases Decrepify and Nevermove's CC. The health and AP on it make it a match made in heaven for Swain. I sometimes skip it if I need more AP however. Also, I prefer to purchase Giant's Belt first, but it's understandable if you don't have the gold for it, put into either amp tome of blasting wand.
Deathcap- Get this item if you are doing really well instead of defensive 3rd. Otherwise, get it later, if the game lasts that long.
Spirit Visage- Another good Swain item. It can be a good mid/early game item, but don't keep it late game. Just remember, this will heal you better but an AP item will heal you better and do more damage. This is an item for, "is good under the right circumstance."
Frozen Heart- If you somehow didn't ban Tryndamere or are versing a lot of AD/AS. then this is your item. CDR and mana help a lot, the more mana the merrier for Swain.
Banshee's Veil- Health, Mana, MR and an amazing passive. I don't think anyone will argue with this, as it's an item almost every champion gets.
Mercury Tread's- Probably the greatest boots in the game. However, Sorc shoes give you that mpen with your runes that allow you to leave MOST enemies at 2-3% MR, which isn't going to do anything unless they stack more. I almost always get Sorc shoes, because I like to build tanky, so MPEN items just aren't as good. Your build is going to give you a solid amount of tankiness, so Sorc shoes should be your choice.
Will of the Ancients- I used to try this in combination with Spirit Visage, it's pretty fun. You heal at a ridiculous rate, it's really funny. But in all honesty, get cleanse if you're trying this because ignite will destroy your entire build. You won't be as tanky as you think you are though, and you won't be doing the damage you should be.
Quicksilver Sash- Another good situational item. Not a very big fan still, I would rather get Chalice of Harmony.
Zhonya's Hourglass- What a great item on Swain. I really love getting this when their AD carry/Tryndamere is fed. Just make sure you're in your uber when you shoot the active and you'll heal up while your CD's reset.
Void Staff- There are a few problems I have with this item. First one is only a problem in lower ELO, where the carries/casters/DPSers get little to no MR. My second problem is the diminishing returns with Sorc shoes. If you're not aware, the % pen does not stack well with base MPEN. Late game however, if you get this, stack AP after. No more MPEN is needed, you will do more damage with AP guaranteed. If you choose to stack more, you're hardly anymore pen for your buck.
Haunting Guise- I know everyone hates this item ever since it got nerfed, but it does a decent job against the 52 club. The few champs that have "scaling MR", Garen, Kassadin, Olaf, Akali, etc. There aren't very many but it doesn't give as much diminishing returns and gives a decent health/AP buff. Another mid-game item like Spirit Visage, that's situational.
Mejai's Soulstealer- Please boycott this item and all snowball items. See forums.
Doran's Ring- I thought it might be healthy to explain when to get this item. Get it in early stages before RoA if you havent bought blasting wand or sorc shoes but are back at base. Buy it if you have a blasting wand but cant afford sorc shoes/ vice versa. I average at least 2 a game, just as fillers when I have to go back. Always remember that when you go back to fountain you have to buy something, that's when this item shines. It's also a great item on Swain for obvious reasons. This is your fallback item if you start the game negative. If you get first blooded or getting beat mid/top, buy this. If you are going 0/2, 0/3 in your solo lane, buy 2 or 3. Don't get discouraged because this fall back will put you on even ground. Just play into their mistakes and get fed. I've started out 0/3, everyone has bad games, and finished with an amazing k/d because of this.
Archangel's Staff- Honestly, the only time you should get this item is when the game has dragged on so long, your build is complete and all you have left is your boots and the enemy team is NOT stacking MR. The mana regen is great and so is the AP at that point but defensive or void staff are honestly the way to go at that point. The above circumstance is the only time you should get this.

-Always solo lane. You depend on last hits early game for your mana regen and harass on one target. You will not do as well duo lane.
-Please don't jungle Swain, you're not Phreak so don't try to jungle any champ.
-Don't underestimate your DoT's either.
-Chances are, after you land a nevermove plus e+q combo early game in your solo lane, the enemy will start running. Anticipate this by running towards your opponent and auto attacking. If you have your uber it will be even more effective.
-This was brought to you buy: Miobi (make it or break it)