As far as last hitting is concerned, it really depends from champion to champion, your runes and masteries. For example, with Draven or Ezreal I appear to have little problems last hitting due to their decent early game damage. But as Ashe it can be a little trickier since if you start with the conventional boots and potions rather than a doran's blade, her damage is abysmal at the early levels.
That's from my personal experience, anyhow. At the first levels, I wouldn't worry too much as people (unless they are smurfs) are still finding things out themselves. Play as many champions as you can at least once if you can, you'll get a good feel for what they are capable of and even better, what can cripple them. Very valuable information to have when you're pitted up against them.
Also, wards win games. A surprising amount of people will leave the warding duties solely to the bottom lane support, but if you're top, mid or especially jungling, getting a ward of two when you have a bit of extra coin to spent can go a long way. Other than that, for champion specific information, most builds on mobafire.com give in-depth information on champions by people much better than I ever could hope to accomplish.
A lot of it truly is just getting to known the champions, who works well with whom, what champion can counter another, things like that. Oh, the guides for jungle champions usually contain maps with excellent positions to place your maps. Oh and another thing, once you get to know the items a bit better, be flexible in your build concerning the situation. I don't know about the American side of things, but on the Europe servers it surprised me how under appreciated a quick silver sash is.
Well, I'll stop creating a wall of text now, these are the things that helped me personally and I usually used to help friends. I am definitely not an expert though, so take any future suggestions and this I might give with a grain of salt.