I would recommend a few things, in combination or individually.
First, The 100 Game Challenge. Developed by the kids over at lowelo.com, the 100 game challenge is a bit of a struggle but if you can get through it can take your game to a new level. They can explain it best over at their site, but I'll give you a quick rundown. Essentially, the 100 game challenge is precisely what it sounds like: 100 games with the same champion, with each block of 25 games focusing on a different aspect of the game.
Games 1 - 25, you focus on last hitting. The goal is not to win the game, or to win your lane, or even not to die. Your only focus is learning precisely when your champion of choice can take advantage of a minion that's about to die.
Games 26 - 50, the goal is item build. Again, victory is not your primary goal. Play around with itemization. While mobafire//lolpros guides can be an excellent place to start, it's important that you not allow them to be the be-all and end-all of your item strategy. To that end, try as many different builds as you can -- a different one each game is ideal. Do your best to figure out what items are so effective they're must-buys, which items are good in some situations and not others, and which you should avoid entirely.
Games 51 - 75 focus on teamfighting. The broad strokes are the same as above, do your best to focus on teamfighting, using the last-hitting skills you built over the first 25 games to build the items from the next set, without beating yourself up too much over your failures. Personally, I find the initiation to be the most important part of teamfighting (with my champion, when should I initiate? Who are my best targets? Should I initiate at all, or leave it up to the Malphite?) but positioning and focus are also vital. For ADCs, teamfights are more about knowing who to hit with what when, whereas for mids keeping yourself safe is usually paramount, and most tops have to worry a bit more about keeping their allies safe than dealing damage.
The final set of games, 76 - 100, is just putting it all together. By now you should have a good picture of what you're meant to be doing during each phase of the game, and during this set start exploring higher concepts, such as what your strong and weak matchups are, your overall style of play, and the other ineffable sorts of things that none the less matter quite a bit in League.
While the benefits of the 100 Game Challenge are manyfold, one of the most important things I took away from it is that you are the most important aspect of your success or failure. While you can absolutely get carried to victory on the shoulders of stronger team members, and get dragged down by wumblers, over a long enough time frame the only constant is you. Focus in on one portion of the game at a time, and it becomes much more digestible.
Unfortunately, the 100 Game Challenge only really works for Top, ADC, and Support. Lowelo has talked about putting out similar things for junglers and supports, but as of this writing they haven't.
Secondly, play Ezreal. He might not be the very best champion, and he might not mesh particularly well with your personal style, but I'm a big believer in newbies playing at least 10 or 20 games with him. Why? Every single one of his abilities is a skillshot, that is, an ability that fires in a particular path, not according to a specific target. Not every champion has a skillshot in their kit, but the vast majority of them have at least one. Forcing yourself to play a champion where you have very little choice but to get good at skillshots can have a dramatic effect on your ability to actually land the dang things.
Finally, Read. A lot. I said above you shouldn't rely too heavily on Mobafire for your exact loadouts, and I stand by that, but they can be extremely useful for a number of reasons. Most guides have the subject's skills laid out in plain English rather than the rather dense and opaque language that Riot tends to use, as well as including some nice bits about how those skills are used. Likewise, most guides have some portion of the text dedicated to why the items highlighted in the guide are used. It can be very useful to have an understanding not only of what champions actually do, but how they are most frequently played.
In addition to Mobafire, lowelo.com, reignofgaming.net, and Riot's website have pretty frequent and comprehensive breakdowns of existing mechanics, as well as upcoming ones.