How to renovate the garage of a house to a game room
In the U.S.:
First, check local building codes and ordinances. Find out what ICC codes (IRC, electrical, fire, etc) might apply, if the governing authorities will require plans or have anything specifically against this (for example, HOAs may require that autos be stored inside, which might ruin your plans pretty quickly)!
Second, get a designer to draw plans for what you want done. Unless you have done it before, do not go looking for subs with a sketch on a napkin! Lack of detail makes contractors nervous, and they charge more to cover unknowns when they're nervous. It's worth a few hundred $$$ to get a designer to draw up a plan - not only to quantify the scope of work but also to make sure all the stuff you want to fit inside will.
Third, find a reputable contractor who is licensed and bonded to do the work. If you can find several, ask for estimates from them all with the exact same scope of work included and described. Make sure that they are responsible for pulling all permits- it will save you a ton of heartache and pulling permits yourself probably won't save much money if any. At the very least you'll probably need the floor leveled, the garage doors framed in (optional), and if the garage is unfinished, insulate/soundproof, hang/tapenade float/ paint sheetrock. If you will be doing electronic gaming, make sure that you have enough electric and data outlets in the right places to hook everything up correctly.
Hope this helps. Feel free to ask follow-up questions!
Generally the first and very important place to inquire is your local building department which should be a part of you local government offices and their information is the most reliable you can get and is typically absolutely free.