- Don't Feed. Feeding is dying unnecessarily and thereby giving the enemy money, turning a good opponent into an unstoppable beast. Don't worry if you don't know what I mean - the rest of the guide will help you learn to avoid feeding the enemy team (and spending too much time dead). This leads me to #2 ...
- Play defensively if you're new to the game! This is one of the biggest mistakes newbies make. It is far better to stay back near your turrets and farm gold carefully (in the early game) than to run ahead of your lines and die. It's better to have few or no kills early game but NOT die.
Choose an easy champion. When you look at a champion's profile, the "difficulty" bar tells you how hard it is to play that champion. Don't ignore it! As a newbie, you will enjoy the game a lot more if you follow the difficulty guidelines. More difficult characters will require more depth of game knowledge to get kills and to generally do okay.
Choose a ranged champion. Some people may disagree, but generally the easiest characters to play are ranged. Ashe, Annie and Sivir come to mind.
They allow you to stand far back from your opponent and harass them from a distance. There are some easier melee Champions like Nunu that may also appeal to new players.
Start the game in a lane with someone else. There are three lanes and 5 players. Someone will have to solo. If you're new to the game, you do NOT want to solo It gives faster levels (because shared lanes mean shared XP) but unless you're good with your champion, it's not going to be fun.
PLAY THE TUTORIAL. You heard me. PLAY THE TUTORIAL! It's not fair to come into a public game without any clue what you're doing. Even if you've played DoTA, it's worth a quick tutorial game. It will get you familiar with everything from basic movement to using brush to stealth (a feature not in DoTA), playing a lane, pushing a lane, etc.
ITEMS - Start with a mana-regen item. Some may argue that this won't work for every hero, and won't work for every play style. That is true. But for a newbie who is going to be playing defensively, you need to be able to continue to use ranged abilities to harass enemy champions. The Meki Pendent and it's 7 MP/5 is very, very useful in the early game. If you're against a number of magic-using champions, upgrading it to a Chalice is not a bad use of 500G. Beyond that, stick with the RECOMMENDED items for each hero. They are always a good bet, and if you look at them you'll usually notice that some are offensive and some are defensive - choose whichever suits your style. Note that one of the many forms of speed boots are recommended for every Champion, and it is strongly advised that you upgrade your movement speed early.
Understand TURRET AGGRO. One of the biggest mistakes newbies make (which almost always results in a death) is failing to understand turret aggro - the order in which enemy turrets choose their targets. The important thing to know is that IF YOU ATTACK AN ENEMY HERO THE TURRET WILL IMMEDIATELY TURN ON YOU. Unless an enemy has only a sliver of health, and you have full health and a lot of confidence, DONT ATTACK AN ENEMY BY THEIR TURRET! (Sure, for intermediate players there are many ways to attack a hero by their turret, but until you figure the game out, DONT DO IT!)
Use your creep. (Creep are the little soldiers that mindlessly continue to spawn in waves, make their way to the front, and fight enemy creep/turrets/Champions) Stay with your creep. Never, ever push beyond your creep in the early game. Your creep do a few things for you. First, they create impediments to ranged abilities like Ashe's Volley. Standing behind your lines makes Ashe less able to freely pick away at your health with Volley, for example. Creep also causes collision, meaning if your enemy wants to attack you, they have to walk AROUND your creep. If you're playing a ranged hero and an enemy is melee, standing behind your creep is a great way to keep them from hitting you.
Be aware of the big picture. This is the least newbie of all of these tips, but nevertheless something you should start to consider from the start. Are all 5 enemy heroes visible on the map? If not, do you know where they are? What part of the map they are likely on? Pretty early on, some players will start to try to set up ganks (that is, a planned kill). Be aware of the map. If enemy heroes are missing, be on the defensive side, closer to your turrets. A useful quote from the feedback - "Never cross the river alone" (Doomenstein, thanks!)
Clearly everyone will have their own play style. These tips are my own opinion on how a new player might improve so that they will enjoy their early games while they learn the ropes. Also, new players don't be discouraged if you lose quite a bit at first. Every time you lose, your hidden ELO score gets lower, and your next match will be against easier opponents. If you ever played WoW in season 5 or earlier, it is like starting arena at 1500. In LoL, however, the blow is softened because if you're Summoner Level 1, it will put you against other players of lower levels.
Good luck. Have fun. Don't Feed!