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I never really had problems with leavers, but I probably just got lucky. IF you climb in ELO rating you will start to see less of them. Unranked ELO is invisible, but you start at 1200, and according to lolbase.net I'm around 1530 ELO and I basically see a leaver once every 25 games.
You should also know that these types of games aren't very new player friendly. That's not to say the community is awful; and in my opinion of the three main games LoL has the most pleasant community. But the nature of the game means there isn't really a training zone or a chance to learn without it affecting gameplay. You kind of have to learn on your feet.
A few general tips for someone who has never played these types of games before.
1. Know where your enemies and allies are. If several of your enemies are missing and you push too far alone, then you will probably get ganked. Pushing alone too far in general is a bad idea.
2. Buy boots sometime early in the game. They probably shouldn't be your first item, but a second or third definitely. It's a huge mistake to get them much later.
3. Don't over-specialize with your items. Ashe is one of the champions on free rotation at the moment. She is best built as an attack damage type character (getting damage items, attack speed, critical hit chance, armor penetration, and more). However you should still get at least one defensive item on her, otherwise she will often be too frail to stand up long in the fight. You shouldn't be a glass cannon, you should be a tempered-glass cannon. Likewise with tanky champions like shen you shouldn't be purely a brick wall, you should be a spiky brick wall.
4. Last hit the minions. You only get gold from the kill if you deliver the killing blow. You gain some gold over time regardless, but it is very little. Most of your gold will come from killing minions, towers, and champions.