What kind of riding do you want to do? How much do you have to spend? So many variables. Go test ride all sorts of bikes and you will be able to figure it out.
I can't tell you what bike to get, theres so many out there, but I would recomend a size small frame, a medium at the biggest but that would still be pushing it.
since your only 5' 3", i'm assuming your not that heavy so you probably won't need a bike with huge amounts of travel such as an M3 or V-10. but as the other guys said, more info would help.
Well your 5'3 right. I'm guessing you're pretty light as well then. Stick to something lighter. Don't go all out on a super expensive bike right off the bat. There's some nice female specific bikes out there. If you want a full suspension try to keep in around that 35lbs weight. Anything that gets up towards that 40lbs range is going to be really heavy for you.
I would look at a Giant Reign. They might be a tad over your budget but are well worth the extra money. I am sure you can find one used for about $1500 but new, the lower end model, the Reign X2 is still very well spec'ed out for what you are paying. I am not too sure on the exact price of that bike but I would assume it is about $2000ish. They also come in an XS
Careful buying online too. Get something close to you and take it to a bike shop before you pay for it. They'll look over it for you and see if there's any money that you need to sink into it. All you need is to spend you $1500 on a bike then spend another $500 on a new drivetrain and break cables.