Some stuff I've learned in two weeks of playing:
If you get an enemy down in health, and they run away, don't follow them into their tower. You will die. The rest of this post isn't as important as this. one. thing.
Be a nice person
Don't taunt enemy players if you're doing better. Don't feed the trolls- if someone is on your case, ask them to stop and then ignore them. Learn the difference between "on your case" and "giving you advice". "Stop doing X" or "You shouldn't take X spell" is probably the latter. If someone accuses you of feeding and you have more kills than deaths it's probably safe to ignore them, otherwise you may want to be less aggressive or switch lanes with a better matched player. If a player really offends you, you can right-click their name and report them for abusive behavior at the end of the game. Equally important- if you see someone communicating well or giving you tips, you might want to add them to your friends list!
There's probably a much deeper code to this, but generally, if the enemy team is attacking the tower, ping it! To ping a building, press g and click on it. I like to send a number of pings equal to the number of players attacking it, but just one is always sufficient. Say why you're pinging whenever you get a chance afterwards, even if it's just "help (lane)!" If you get killed by 3 or more enemies working together, you may also want to ping where you died. If your lane opponent wanders off into the forest or BPs, say "mia (your lane)", because they might just pop up and try to attack another lane or get involved in a teamfight or gank. When you first meet your lane opponent(s) I like to call which characters they are.
Don't initiate early
Don't get in close and start PVP fights early game. Whip your ranged abilities at them and try to do a little damage, but try to avoid them also damaging you! This will mean they have to leave to heal first. The rest of the game is pretty much yours if you can do this well.
Using your towers
The best way to get kills early game, is to trick the enemy into walking into your tower. This is called kiting, and it's much easier to do if you're playing a champion with ranged attacks. Just like you, when they get hit by something and aren't doing anything else, they will move closer to try and retaliate- automatically. If they're close to your tower, use this to your advantage. If they are injured or not paying attention you might net yourself a First Blood! After reading this you know better then to chase an injured enemy into his tower and die, but does the enemy know better? If you're hurt badly, GET TO THE TOWAH.
If the enemy is attacking your tower, try to push them back! If you can't get them to attack you, killing all the minions will force them to retreat. If they try to stop you, the tower will shoot them! A bad player will be killed and a good player will be forced to run away.
If the enemy kills your tower, it's gone for good! Retreat to the next one! This means they've got one less obstacle, but it also means they have to run farther to get back to the safety of their tower! There's no way to repair towers, so defend them! An exception is the last two towers by your Nexus. These towers slowly regain HP as long as they aren't destroyed completely. You should defend these even more fiercely- if they go down you've almost surely lost!
Dealing with enemy towers mid and late game
Around mid-game, if you get your opponent out of lane, you might want to soften the tower a bit. Listen for the announcer to say one of your allies has taken down a turret- this is probably a good time to start going for the tower in your lane. The tower will focus the first thing it sees, and then will only attack minions until they are all gone. Most abilities will not target or damage turrets. If you're playing a mage with weak physical attacks it may be best to keep the enemy minions back with your abilities so yours can focus on the tower instead of attacking it yourself. Watch your minions- when there's only one left, run!
Watch your minimap. Learn which enemies are in which lanes, and which of your allies are facing them. If you're not soling your lane, and they are getting rolled, it might be good to go and help, if you can get there fast enough.
Choose wisely. Middle is often a solo lane early on, reserved for the carry who needs to build himself up with gold. It also has frequent teamfights late game. Top and bottom are symmetrical, and which you go on will often depend on where the rest of your team goes. Your lane opponents are important- if you're a melee champ who relies on movement like Singed, you don't want to be fighting against two people with stuns or slows. Ask someone to switch with you and give your reasons- but be quick about it, don't stop and type a paragraph when you're alive.
Don't push the lane early game. If you are moving forward before their next wave of minions come you're pushing too hard. Try to only attack to lasthit for extra gold. Don't worry too much if you have a hard time, you'll get it as you play that champ more. Later in the game, if you get an item that increases in power such as Manamune, you might want to attack/use abilities anyway, but don't run out of mana or push into their tower if you can help it!
Remember the jungle and brush! Expect to be attacked from it. Remember that you can flee into it instead of down your lane, interrupting your enemy's line of sight with brush and corners as you go, and pop out next to your tower. Don't put brush behind you without checking it. If the enemy is out of your fog of war and you're not by an enemy tower or minion, they can't see you! Now is the time to get into brush. Even if you hardly damage them, the threat of a gank can make people retreat even if it's a bluff. If your teammate is engaging an enemy and you have CC abilities, it might be wiser to go around through the jungle and attack from behind to cut off their escape, but don't challenge their tower to do it!
It is sometimes necessary to leave your lane to help your team. Watch for chat and pings. If there is a teamfight, go help! If they are trying to destroy a tower in another lane, go help! Another good time to leave your lane is if your lane opponent is 5+ levels below you- it may be more helpful to your team to "gank"(surprise attack) a more evenly matches enemy in the middle lane or jungle. You may not want to do this, but I also like to go do something else if it looks like my lane opponent isn't having any fun or is too far below my level, to be sporting.
Learning the game
Watch when the enemy attacks to learn how their abilities work and what they look like. Not only so you can counter them, but because you may like their abilities and want to try them. Learning to recognize attacks is important too- Some champs like Janna, Veigar, Pantheon, Lux, for example, have powerful but delayed skills that you can see coming(known as "skillshots") so you can often get out of the way(or, occasionally, IN the way, for example if you're healthy and see Ashe's Enchanted Ice Arrow coming you might want to eat it to save an injured teammate behind you) Summoner spells such as Heal also have unique special effects that are visible to all- remember what they look like so you can recognize them.
Use potions early game! 200HP is huge, and can save you from having to bluepill out and miss experience.
Get boots every game, sooner rather than later. I like to get the first boots after I get two of the ingredients for my first core item, and then upgrade them after I finish it, but you may need them sooner. Boots are a good way to counter the enemy mid-game, it's often a good idea to pick ones that counter the enemy team if you're having trouble.
Pay attention to your enemy- if you left click a tower or minion you can see their health, if you click an enemy champ you can see their items and stats! If you don't recognize an item, read its description to know what you're up against.
This is also important after the game. Look at each player's items, their score, think about what they did that was effective or ineffective. Life is too short to learn from only your own actions, so try to learn from theirs as well!
Look up guides for your character. Good sites to find guides are Leaguecraft and Mobafire-But LoL is popular enough that you can just google "character name builds" and get plenty of results. Don't follow any one exactly(you won't be able to because of runes and masteries), but get a feel for what kinds of items and which stats are best, and look for tips on using your abilities effectively.
You can look up guides before and after your first rounds with the character. If you look up guides first, you will come in with a better understanding, but some things you will learn faster by laying the champion than reading about them- especially if the enemy has someone playing the same champion. If they do, watch them whenever you can. Sometimes guides will make alot more sense after you've played your first round with a character.
This changed in the last patch, but might change again in the future so I'll say it anyway. Custom games against bots used to be a sure way to get some decent IP every day and practice new champions. Now, if you're new enough to need this guide, the bots will probably be stronger than the average human opponent in ranked games and you should just play those, since they are worth more. You can only get IP and XP from 4 custom games per day.
Someday soon they will be adding a feature called "the tribunal". You will be allowed to view cases of reported abuse, harassment or griefing and vote on whether a punishment is warranted or not for the reported player. You will be given some IP for each case that is decided the same way you voted, as a reward for helping ease the burden on the moderators.
Pay attention to the "first win of the day" timer- you get a good extra chunk of IP for it. Note this is YOUR first win of "the day". You can get it any time as long as, at the end of the game, it's been at least 24 hours since your last first win of the day bonus, and you're not playing a custom game.
Try different free characters, especially ones that are similar to one you're already good at. Try all the free week characters each week if you can. If you have several good games with one, you might want to buy it to use when they aren't free next week. You only need to be good with a few champions at this point. You can expand your repertoire later when you've got full runes and masteries available!
Contrary to what OP said, some level 1 runes can help. For example blue runes of intellect(flat mana), red runes of Insight(flat magic penetration) or alacrity(flat attack speed), or yellow runes of resillience(flat armor) are a safe investment for just 30 IP each- less than two good tier three runes for 18 of them. Do NOT buy level 2 runes, however. Wait until level 21 when you can buy level 3s. It will cost thousands of IP to outfit your character with a full page of level 3 runes. So start saving! Don't buy champs all willy-nilly.
You only have to buy, at most, 9 of any given rune. Once you have them, you can use them on any of your rune pages in any combinations you want without buying more of the same rune.
There are three types and three colors of runes, Primary, secondary and quintessences- primary are strong runes that fit the color they are, secondary are much weaker and don't fit their color. Examples of red primary runes include damage, armor penetration and critical chances, yellows include health regen, armor, dodge, and magic resist, blue incudes mana, ability power and cooldown reduction. Quints are never secondary, and go in special slots. Avoid secondary runes if possible. You should almost always get all of the same type of rune in a color's slots to maximize the effects.