Here's that actual information post I promised. I don't really know anything about you or your playstyles and weaknesses, so here's a general page about supporting that I just puked out on my keyboard. I'd be happy to offer more specific, constructive advice given a little more information or targeted questions. This post is from the view of an unranked support main who started playing a little over a year ago, so use appropriate skepticism and compare it to other advice before taking anything to heart. Also, I apologize if this is a little dry. My usual humor was mostly absent when writing it up.
Definition of a support champion: Support champions are defined by kits that allow them to be effective even without the gold and experience taken from farming, usually featuring utility abilities that do not rely on scaling. Supports generally share bot lane with a marksman ADC and an opposing marksman/support duo, but they can aid allies in any position as strategically necessary. We will focus on the standard marksman/support duo lane format.
The support's general role in the game is to help their damage dealers do damage without dying. This can be accomplished through vision control, peeling, initiation, disengage, healing/shields/buffs/debuffs/debuff removal, and many other methods. Strategically, supports should emphasize map awareness and awareness of enemy positions, diplomatically directing their team in avoiding risks and maintaining map-wide vision and objective control.
In lane: The ADC builds late game sustained damage for team fights and objective control, but is extremely vulnerable due to low defensive stats and few escape mechanisms. The support's job is to help them survive the laning phase and get enough gold and experience to be relevant come mid game. As a second priorty, supports work with their ADCs to deny their lane opponents gold and experience. In other words, the role of the support is to help their ADC get fed while keeping their opposing ADC from getting fed. What does this mean for you?
First priority: Prevent your ADC from dying, and don't die yourself unless your death will save a valuable teammate (not all teammates or ADCs are worth saving). Don't take any damage you don't have to take. Learn to dodge skillshots and harass when enemies are on cooldown.
Second priority: Make sure your ADC can farm safely. Ward against ganks and keep opponents off the ADC. Ping caution and recommend wave management patterns (push, freeze, retreat) as threats come and go. Use your auxillary ward (your ADC's yellow trinket) to ward in the lane and in nearby bushes. Build a gold item and use it as often as possible to keep yourself equipped, even if you are not last-hitting when allies are in the lane. If you can, avoid last-hitting enemy champions so your ADC can get the kill reward, but know that securing the kill is more important than trading assist and kill credit.
Third priority: Work with your ADC to secure team objectives. Push the bot lane tower when your lane opponents are absent or weak and you are safe from mid/jungle ganks (It is fine to help push, but don't start attacking minions unless your lane partner knows what you are doing. ADCs focus hard on wave management, and may not take kindly if they think you are interfering with it.). The sooner you capture the bot lane tower, the sooner you can roam and push other lanes. Build your sightstone and sweeper early and use it to ward against jungle ganks and to keep an eye on dragon. When your team gets an advantage in bot (when bot lane is dead and the jungler/mid is dead/in top and bot is pushed), take dragon, warding enemy paths of approach before doing so to prevent a steal or unfavorable fight. Do not let your silly jungler solo dragon without wards, and don't sacrifice yourself to save him when he does.
Fourth priority: Deny the enemy ADC farm. Work with your ADC to harass your opponents to keep them from farming or engaging, or all-in and kill them if the opportunity presents itself (this is risky, but quite effective for denying farm). Note that this is the last priority in this list. Remember that games are won by capturing objectives, not by killing or damaging opponents, and trading with opponents always has an element of risk attached to it.
Your choice of champion and behavior in lane should be adjusted according to your ADC's champion and playstyle, and to your lane opponents. Talk to your ADCs to decide on and adjust your approach to laning. As general advice, I would recommend learning the marksman ADC role, as knowledge and experience in the role will equip you for better supporting.
A few things I consciously work on to make myself a better support are mana management, safe harassment, and warding according to the enemy team. Every support player can improve on damage and death avoidance in all stages of the game.
Post-laning: Wave management, objective control, and teamfighting/split pushing define the middle and late game. In solo queue, your team will probably not have much coordination, so do your best to guide your team, focusing on: Staying out of dangerous (unwarded) areas (especially when chasing), coordinating team vision control around contested areas (ask people to buy pinks and upgrade trinkets), and split lane pressure and rotations to capture objectives without strong opposition. Strong vision control will allow you to avoid being caught out (especially when split pushing) and will help you make picks on enemies when they are out of position, so they have one less man/woman/plant for defending objectives. Unless all of your teammates are neglecting side lane pressure, you will probably want to stay near the core of your team where the most serious fights will break out, even if your ADC is managing a wave in a side lane. If one of your immobile carries is pushing far into enemy territory and is not warding like they should be, it is fine to try to ward for them and stay on hand to help them escape.
Your behavior in fights will vary significantly based on your champion, current build, and team composition, but your essential goal will always be the same: Help your damage dealers do damage without dying.
Pick supports (Blitzcrank, Thresh) specialize in punishing out of position enemies with instant death, turning 5v5s into 4v5s. Use stealth and surprise to your advantage to land your hooks, and always make sure you have sufficient follow up to turn your grab into a kill. Aggressive ward placement can seriously improve your picks, so long as your team has enough defensive vision to be able to afford it. Clear enemy wards frequently to guarantee you will not be seen when sneaking a grab.
Peel supports (Janna, also Thresh, Alistar, Nautilus) and their counterparts in the other lanes use crowd control to prevent enemies from reaching your damage dealers. Some, like Taric, excel at counter-engaging and consider enemy dives to be excellent opportunities for a pentakill. In general, peel supports should stay near their damage dealers so they can keep enemy front liners from diving them.
Mage supports (Vel'Koz, Annie): Let's be honest, you played these champs because you wanted to be an APC but your team needed a support. Land huge stuns and do crap tons of damage, just make sure you are keeping up on your other support duties. (I don't play these and generally don't like the class, so don't take me seriously as I have no idea what I am talking about.)
Engage/disruption supports (Leona, Thressshhh, Nautilus, aggressive Jannas, Sejuani) specialize in making the entire enemy team sit down and think about their life choices while your ADC/APC/Taric grind them into a fine dust. Your job is to get into the middle of the enemy team and keep them from doing anything more aggressive than starting "Nerf Nautilus" threads on the boards. Be very sure that your team is ready to follow up, as diving an enemy team without a serious external damage source will do nothing but guarantee your humiliating demise.
Healing/shielding/poking supports (Soraka, Lulu, Sona, Janna again) will generally stick to the back lines of combat and throw damage/light CC into the enemy team while healing/shielding front liners and damage dealers as necessary. These tend to be very squishy champions, so learn how to keep yourself alive and weigh the risks of trying to save over-aggressive allies. If you want to improve damage output, an Ardent Censer on supports with distributed or frequent heals and shields can make your AA-based allies much more intimidating.
A final note: Sometimes the most important thing a support brings is a good attitude. Be friendly to your allies and opponents, complement both sides on excellent play, and generally be a pleasant person to have around. Even if you are playing in Ranked, League is a game and should be enjoyed. Do your part to ensure that everyone is enjoying the games you play in and you will have a better time for it. Feel free to ask me questions, call me out on my inaccuracies and foolish content, or add me in-game (if you are not the poster and want to add me, leave a comment so I know you aren't a spam bot).