Hey PG, glad to see another young person getting in to the sailing lifestyle. Many good points have already been made. I will try to give a few of my own insights, perhaps they will prove useful.
I myself am 24 years old and own a 27' sailboat. I paid $2900 for her and have put about 5K into her with another 2K being needed for completion. I bought a project boat and what a project she has been! 10 months later and I am four weeks from moving aboard.
Your budget is plenty. I would look at boats in the 10-15K range which leaves plenty of money to do your own refit as well as the SURE TO BE money stealers you will find aboard your new home.
If you are mechanically inclined good for you. You will save loads of money by doing your own work.
Your biggest worry/concern is going to be getting your girlfriend to enjoy the lifestyle to the same extent as you. Boats are cramped quarters so everyone should be on the same page.
Also, where you are planning on staying is cold and wet. Read into condensation and winter wrapping, those will both be important for you and your comfort aboard.
As far as boats go? Stay below 35' feet. Look for production boats like O'Days, Morgans, Pearsons, Cheoy Lee's and so forth. You want to enjoy being a live aboard not being broke. I recommend you avoid the Rasseys and seacrafts for a later date.