Hi Emily, Welcome!
I recommend you go to this thread:-
http://www.tarotforum.net/showthread.php?t=15119
It is a table of contents for reading the cards. If you scroll down quite a bit there are some good threads there, including others just like this one - "advice for beginners' and 'learning the cards' for example in post #6.
As for individual stories, well, I'm still learning, but I guess the way I started was to read the book. I was lucky, my deck came with a proper book with a double page for each card, explaining the symbolism in the cards as well as meanings for them.
As I developed, and I guess this is the stage I'm up to, I 'know' most of the card meanings, but for the most part I 'guess'. Lol, it is called intuition, but sometimes I have to say it still feels like guessing! I am learning to trust myself more, because I'm usually right. I look at the cards and try and think of what they are telling me. Sometimes, to be honest, I think I confuse the book meanings on cards, get them mixed up, but that hasn't seemed to matter yet. WHat I look for in the cards is what I interpret the situation to be happening, as well as the expression on the people's faces. The expression is what usually gives me a clue to how the card should be interpreted - is the person happy, sad, excited, no emotion? Then the expression to how it relates to the card - there may be some kind of stuggle going on, but the main person looks happy, so the meaning to me is that the person is enjoying a bit of conflict, or maybe the person needs to lighten up about the conflict and see the positive side, etc.
I also for the moment add in what I interpret the suits to mean - as in wands are creative energy, cups are emotions.
I guess I make a story out of the card and the elements, and this is how I read. Other people do it different ways, but this way works for me.
I also note if something stands out - a colour, a shape, an animal. If these things are drawing your eyes then they are probably important.
As for how long before I became familiar? Not too long I guess. I started at the end of 2004, and I practised alot, I think I was fairly familiar in a month or two later. I've had periods where i haven't touched my cards for months, but when I pick them up again I can read with them just fine. I practised alot in the second half of last year and I think that is when I finally started to 'read' not just half read the cards, half read the books.
And now, a year and a half since I began, I really feel more confident with myself and my reading. I think that is a big battle for some people - trusting yourself and your intuition.
Hope that has helped a bit. Good luck!