Some stretches to do after your pups:

Chest stretches (each link has instructions and illustration)
Anterior deltoid stretches; push your chest out while doing this one to deepen the stretch in the chest muscles
Overhead triceps stretch

Make sure you're getting enough protein. We're building muscle, so we'll need extra.

Expect to be sore... a LITTLE sore... for pretty much the whole program. If you're so sore that it interferes with your other activities, that's too much, and try taking an extra rest day and/or eating more protein. But really, with body weight exercises and a limit on the number of sets and sessions, that shouldn't happen after the first day, if at all.

If you start getting joint pain, that's not good. Yes it's tough to drop out of a program because your body won't do it, but that's better than keeping on and aggravating an injury that could take months to heal. If you can back off the number of reps and/or do one of the modified types of pups, and do them without ANY joint pain (during or after the workout), do that, give yourself a week and then try increasing the reps again. Have someone check your form, and/or use a mirror. If it's still painful, then STOP. If you start getting weird but non-painful things going on in your joints - noise, clicking, crackling, etc. - pay attention to that as well and don't let it develop into something painful.