I don't know what level of advice you need, so here's some really basic stuff. Let me know if you have questions or more specific concerns, and feel free to add me. I enjoy playing with and teaching new players, so I'd be happy to work with you, or just be a friendly guy to fill your team builder roster. I'm a support main who often fills as a jungler or ADC, so my advice on bot lane roles will be more reliable than that regarding top and mid. Keep in mind, League is a complex game, with many champions, strategies, and items. You won't be able to learn everything at once, so don't beat yourself up over it if you don't understand everything (I sure don't!).
Rule one is DON'T DIE. Obviously there are exceptions 
, but it's a solid general rule of thumb, and your games will typically go better if you avoid unnecessary risks, like engaging when you are out-numbered or chasing an enemy into unwarded territory. Not dying is very important in every position, but I would argue that it is most important in top and bot. Remember that your first priority is capturing objectives, not necessarily killing your enemy. Don't risk uncertain fights unless you have to.
Try to avoid taking free damage. Minions and towers both follow the same attack orders: Attack the first thing that comes in range of you, then prioritize minions if that thing leaves range or dies. However, bot will start attacking you if you attack one of their allied champions while in their attack ranges. Towers always do a lot of damage, and minions are quite capable of killing you at lower levels, so try to avoid fighting the enemy in their own minion wave or under turret unless you are sure you can kill them and get out alive.
Experience, gold, items, statistics and scaling: Apart from your simple mechanical skill and decision making, your champion's power is determined by your gold and experience.
Experience allows you to increase your champion level, gaining improved abilities and base statistics. You gain experience by being near enemy deaths, usually that of minions. This is why the jungler and support tend to be under-leveled; they both spend a fair amount of their time running around away from battle, either warding or setting up ganks. Even in lane, you can be denied experience simply by standing too far away from minion deaths, but this isn't a concern unless you are being zoned out really hard.
Gold allows you to buy items, which grant stat bonuses as well as passive and activated abilities. Unless you are a support with a gold item (
), most of your gold will come from last-hitting. When an enemy or monster dies, only the person who dealt the killing blow is awarded gold, though assists do sometimes grant partial gold, as with champion kills. This is why last-hitting is important. I recommend you use custom and AI games to practice farming gold from minions by dealing them the killing blow, rather than just constantly attacking them. As a support, you should plan to take a gold item and use its specific gold generation mechanism to get gold. This means that you should avoid last-hitting minions (and usually champions, if you can help it) when an ally without a gold item is present, usually your ADC.
Once you have gold and experience, you need to know how to use it. Your base stats will automatically improve as you level up, but you will have to choose the order in which you level your abilities. In general, you will want to level all of your basic abilities (Q,W,E) at least once by level 6, and level your ultimate (R) at every opportunity (levels 6, 11, 16). Once you have a point in each ability, it is usually wise to maximize one skill before leveling up the others. This is not necessarily your most useful skill, but rather the skill that will be most useful to maximize. Pull up the League wiki page for your champ (ex: http://leagueoflegends.wikia.com/wiki/Soraka) and look at how cooldowns, mana costs, damage, and side effects change as each ability is leveled. On a champion like Soraka, you will usually maximize her E last, since its primary use is utility (silence and root) which is not significantly affected by leveling up, whereas her heal and Starcall rely on number improvements to be useful. If I wanted to poke in-lane, I would max Q first, then W, then E. If I wanted to sustain, I would max W, then Q, then E.
Note that most abilities have some form of scaling. That is, their effects are increased based on one or more of your champion statistics, usually AD or AP. Every champion has a base AD stat, since AD is your auto attack base damage, but you will not have AP unless you buy items that grant it. Champions also have base movement speed, armor (reduces physical damage), magic resist (reduces magic damage), and health. Ability scaling stats will typically be a good indicator of what kind of items you should buy, but item choices will also rely on situational decisions (buy armor if you keep dying to physical damage, build armor penetration if enemy armor is keeping you from killing them) and your role on the team. Be sure to stay aware of how much gold you are carrying, and take some time to base and buy if you are carting around more than two thousand or so, as beating your enemies with a 
is much easier than beating them with a sack of cash. I'll make a reply post with information about champion roles and how they influence build paths.
You will apply the power gained from items and levels to capturing objectives, and, if convenient or necessary, killing your opponents. Objectives include structures (towers, inhibitors, and the nexus) as well as Baron Nashor and Dragon. Structures are the most important targets in the game, as victory and defeat is defined by the destruction of the losing team's nexus. You need not destroy all enemy structures to win, but destroying structures in each lane gives you more freedom to move safely around the map, and can force your enemy to spread their defense across multiple lanes, making the capture of specific objectives easier. Destroying a structure gives everyone on your team, living or dead, reward gold regardless of whether or not they contributed to its destruction. Most damaging abilities do not affect structures, so don't waste your mana on them. Instead, use auto-attacks to break them down. Quickly destroying structures is one of the most important parts of an ADC's job.
Baron Nashor and Dragon, though not essential, are important objectives worth contesting. The first four times your team kills the Dragon, you all gain a permanent stat bonus. Thus, reliably capturing Dragon can significantly improve your late game threat, and denying Dragon kills to your enemies can set them behind. The fifth Dragon kill by your team will give you a large, temporary bonus that can significantly impact the outcome of a team fight. Killing Baron Nashor gives your team a temporary bonus granting an accelerated recall and large bonuses to nearby lane minions useful for breaking sieges and split-pushing. It is recommended that you do not attempt to fight these monsters alone or without enough wards to see enemies coming to stop you well in advance. Junglers should try to keep a charge of smite handy whenever these monsters are alive, either to secure the kill or steal it, since, as with lane minions, only the last hit counts.
Why to kill your opponent and why to not let your opponent kill you:
Dead men earn no (bonus) gold or experience. If you kill your enemy and stay in lane, you will gain a significant lead over them by having more gold and experience. In fact, you get a lot of gold and experience simply by killing them, as if they were an overlarge lane minion.
Dead men do not defend objectives. For some reason, most enemies will try to stop you from destroying their towers or from killing the dragon. The most effective way to prevent this is to kill them.
Dead men cannot dance. Press control 3 or enter the text command /dance to dance. Do not press D to dance unless you enjoy wasting summoner spells.
I'll try to make a second post here about champion roles, the "meta," game phases (laning, mid-game, late-game), and fighting champions. As I mentioned at the top, feel free to ask questions, correct me, or add me if you would like to play. Happy farming!
Edit: The boards seem keen on ruining my formatting, so I apologize if concepts are not arranged spatially as well as they should be.
