You will be away from your wife from the beginning of basic, through OSUT, Airborne, SF prep course, and SFAS. If you get selected, your wife will be able to move out to Bragg if that's what you choose.
Once you start the course you will be gone for 3 weeks of SERE, then home for 12 weeks of MOS, gone for 7 weeks of common skills training, home for a week, gone for 4 weeks of Robin Sage, then home for 6 months of language training.
It is highly unlikely you'll be able to take leave and travel anywhere during the course, besides the 2 weeks of Christmas leave. However, your wife could come travel to see you whenever she wants. You would have to pay for a hotel however, since you'd be in the barracks with a room mate.
I recommend people PCS their family for the course if possible. It makes things a lot easier as you wouldn't have to be in the barracks, and could have a relatively normal home life.
Things change if your a 18D, as the training is much longer.
If you have more questions, search the forum, as this isn't the first time I've answered this question. I'm confident most questions you have, have already been answered.
One more thing. This isn't something you go into half heartily, you need to be all in with no reservations, and your spouse must be the same way, or else you're going to fail.