I changed my last name when I got married. I pretty much just had to get the DMV to give me a new driver's license and Social Security to send me a new card (the marriage license and JOP took care of the legal process for me.)
Edit: now that I think about it, it was kind of a pain in the ass. I had to change my name with insurance, employer, banks, etc. I remember thinking that I should have just kept my maiden name, hehehe.
Anyway, good luck. You have every reason and right to change your name and I think it's great that you're honoring the people who acted as your real parents all these years.