You'll have SUCH a great time...Edinburgh's a fantastic city!
I like the Grassmarket, it's got some good bars and cafes and, as it's more of a student-y area, it tends to be much more relaxed than George Street. However, there are some great bars on and around George Street too...Oloroso is, in my opinion, the best bar in Edinburgh. It's on Castle Street, just off the corner with George Street, and you have to look for it as it's quite unassuming! Dragonfly on West Port, just before the Grassmarket, is a great bar too. Nice and relaxed, GREAT cocktails, and just a nice, cosy size. To be honest, a city like Edinburgh is at it's best when you just go out with an open mind, wander around the streets, and discover the bars yourself. I discovered Oloroso one afternoon, purely by chance, and it's been my favourite ever since.
Enjoy!