How do I play as the Support and get better at it?

 

Don't take abuse from your teammates

First, the advice of others has been pretty good I would say. To add, don't take abuse from your teammates. As a support your job is to be ready to support, with Wards, Smokes, Dusts, Town Portal, etc, but also keep in mind, your job isn't to help your carry who is alone, out of position, has no defensive tower to get to, and is raging at you because he got ganked in the middle of nowhere, when there's no vision available and no enemies on the map. Carries are required to play smart, just as supports are. You can't be blamed for their feed.

The number one thing above all others, is having a Town Portal

Watch the map, if you see your carry or mid getting jumped, Town Portal in, throw a stun, throw your body in the way, do whatever you can to keep them alive. If they're at the tower, then they're in position, they're just in trouble. It's OK if a support dies for a carry to survive, as long as it's a worthy situation. Not to mention, I've had many many many situations where, my mid is getting ganked, I see this, I Town Portal in, and we end up turning the fight.

Finally, never underestimate the value of a smoke gank

Get to level 2/3 grab a Smoke and try to get a mid gank. Wrap around and help secure first blood, or any type of early game kills. The better your team does during the laning phase, the smoother the rest of the game goes. This game thrives on early advantage, and in most situations, a team will snowball off of it. If your mid is 1 or 2 levels above the opposing mid, you will A) get more ganks from the mid because they're ahead and can afford to leave the lane and bully other lanes B) you have teammates that give you some respect and stop spamming you with "we need Wards, Dust, Sentries , etc."

Use Town Portals for critical moments, ie. ganks, tower push, etc

If you die, don't buy a Town Portal just to get back into lane. Just walk. As a support, you're not making a lot of gold, so to waste it just to get into a lane where you'll continue doing not too much, is a waste. That's a set of Wards you could have had, or a Smoke or some dust.

Leave farm for your carries most times

Unless you're doing a hard push, then don't be using your nuke to clear waves. A support should never be saying "but I need farm". Your natural gold income is enough to get a Town Portal about every minute and a half, (100G / minute). Item priority should be Town Portal , Wards, Smoke, (Sentries /Dust when required), courier of course, and then try to at least get your Boots of Speed so you're not a sitting duck during team fights.

Most of your income will come from the natural GPM and then from ganks and towers

Believe me though, as a support, even when you're not farming, if your team is progressing, you'll item up just fine, even while keeping Wards up and having Dust /Smoke etc. The only times I struggle with items as a support are when our team is getting stomped, and then at that point, it doesn't matter much anyway.

Most pubs you'll be a "solo support" which means, it's much harder to get things like Mekansm, Drum of Endurance, Force Staff, Eul's Scepter of Divinity, etc. because techincally there should be a 5 (consumable support items, Wards, Dust , etc.) and a 4 support that buys core support items, such as the items I mentioned above. If you're solo, you can either explain that you're going to be focusing on Wards etc. and you won't be able to get a Mekansm, or, hopefully someone will play a core hero that does well with a Mekansm, like Doom, Viper, Meepo, Necrophos, etc.

Supporting can be fun, but it's very hard and can get stressful, especially as a solo support

You'll be blamed for a lot of things and people will rage about you when they're losing because they think it's the supports job to save them every time they make a retarded play, but, that's when you just mute them, because, if you're doing your job with creating space with ganks and providing vision etc. then the carries should be able to do their jobs with a sound mind that there's no one walking up behind them.

I want to add one more aspect on top of everyone else's advice. It's true that the hard support is supposed to be giving the easy farm and xp to the cores. It's true that for the most part, your life will always be worth trading to save a core. It's true that you are mostly trying to get by with minimal items.

That being said, being support is not an excuse to have terrible farm or levels 

A lot of support players get locked into the "woe is me I'm the martyr" mindset. Just because you are support doesn't mean it's okay to have a KD of 0/15. It doesn't mean that come 30 minutes into the game, you can have nothing but brown boots.

Your job to squeeze as much gold and xp out of the map as possible

You give the easy gold and xp to the cores so they can help carry you late game but it's still your job to squeeze as much gold and xp out of the map as possible. This means setting up kills, jungling, occupying free lanes when possible, whatever it takes.

It is still your responsibility to setup Wards, help ganks, provide Town Portal support and all that good old typical support stuff. But don't think that because you are the "lowly support" your death means nothing and you can have nothing. Watch some pro matches if possible. Sure when someone is getting stomped their supports become dirt poor. But for the most part, Lion, Crystal Maiden, Shadow Shaman, Earthshaker these supports are expected to farm Blink Dagger. Supports need their brown boots even more so than the cores do for mobility in ganks and fights. Having a push stick is a huge benefit for repositioning yourself, your cores, or your enemies. Supports should have a stick if not a wand to provide sustainability in roaming (or some other item like urn, medallion, whatever).

Sure you won't get the big items and you won't be the first to lvl16, but it is important to squeeze those resources out of the map so you can be as relevant as possible. Don't get stuck in the mindset of "oh I'm supposed to have nothing". No you are supposed to be able to pick up those last drops of resources that the cores can't be bothered to grab and use that to maximum effectiveness.