Some games I've played online with the vita that worked well are NFS most wanted, Virtua tennis world tour, FIFA 13, Little big planet and WRC3.
All seemed pretty lag free and some were connected through 3G.
My main gripe which I guess is the same for any console is when people are losing and decide to disconnect. That's the main problem IMO.
I found FIFA 13 servers found games better than when I had FIFA football and Black Ops Declassified just connected to a lobby when it felt like it, even then it only actually loaded the game when it felt like it as well. Shame as I didn't mind the online for it.
As said, trophies sync all under your PS3 user account. If you have PS+ on the PS3 this is also accessible on the vita, if you haven't got PS+, then that's what I'd be recommending as the content gets updated quite frequently and is free/discounted for the vita and PS3.