Most Helpful Girl
Almost all my relationships went "official" when the guy asked me verbally to be his girlfriend. It wasn't necessarily a big deal as in a huge dinner or going to some special place. My one ex asked me to be his girlfriend when we were watching a movie at his place and another ex asked me when we were on his computer 0_o So yeah you basically transition by asking them in person if they'll be your girlfriend. That's it :)1
Most Helpful Guy
Yeah normally you talk about it explicitly, and the dividing line between (unspoken) not-boyfriend-and-girlfriend, and official boyfriend-and-girlfriend, is meaningful and very clear almost all the time.
I would just normally say, "Hey, you wanna be my girlfriend?" or "Are we boyfriend and girlfriend? What do you think?" After it was relatively clear we were a couple. Once I said, "You're my girlfriend, I'm claiming you," because I knew she would like that (and I knew she liked that kind of mild aggression, especially from a caring guy who wouldn't be unkind to her or mistreat her).
It's a fun conversation to have because it's usually quite liberating for both parties. Good luck.0