It always varies. Sometimes you'll pick up the guitar and get warmed up within 5 minutes. Other times you could sit there warming up for an hour and only then start feeling comfortable.

I just do different finger exercises to warm up with. Or go through different scales. I'll focus once on my fretting fingers, then another time specifically on my picking hand.

As far as speeding up legatos. Just practice like everything else. Spend a lot of time just doing legato exercises up and down the neck to a metronome, then increase the tempo slightly each time. Same exact formula as any "How do I get better at ___" question. Practice, practice and practice some more.

edit: and then practice even more