Steve has you covered on the floor with a minimum of 8" joists (spanned appropriately) and braced down to the concrete floor. We are assuming that you have a step down garage and the garage ceiling is the same height as the rest of the house. If not then let us know what you do have. Do you have a basement under the house? What is the distance from garage floor to floor of house?
Steve answered your basic question but there is more before you even start laying out floor joists. You need to have your plumbing, electrical, and heating contractors in to design their so you can work electricial, plumbing, and ductwork in with your joists.
1. Electrical. With what you describe, you are looking at around 8 circuits to cover the master bath, computer room, lighting, hvac, etc. You have to decide where to run all wiring and if your house panel doesn't have 8 (more or less) spaces then where would you install your subpanel.
2. Plumbing. For your master bath, you will need 8" to 12" of room under the floor for traps and sloping of drain pipes. Depending upon where your sewer lines are, you may be looking for a jackhammer in the floor to get them in. They also need vents to the roof.
3. HVAC. You are talking about 625 sq. feet which is about half the size of an average house. I doubt if your furnace and a/c was designed to handle the space in the garage. Depending upon the age of your current equipment and location of ductwork, returns, etc., you are looking at a new single system or adding another stand alone unit somewhere in your garage along with all new returns and ductwork. The floor joists have to be designed to carry all ductwork. And even if you think about a larger furnace and a/c to replace your present one, the main trunks for the ducts and returns may not be designed to handle the extra cubic feet per minute (cfm) for the garage which could leave you with an unsatisfactory system that won't heat and cool all rooms right.
Basically building half a house is an expensive proposition and should be well planned out in advance and I commend you for asking in advance. You should get an architect or draw up some plans.
Other than that, when you have some specific construction questions, post them and we will baby this project along for the rest of the year (or more if you diy it in your spare time). Keep us posted on your progress.