For the core, push ups are good..I would try bench press if you have access to it. Try to keep the bar level whole time.
When I was going hardcore on doing bench press, I started with 20 on each side, do 10~15. Then 25 on each side do 10~15, Then 35 on each side do 10~12. Then do 25 on each side do 10~12.
Initially I could only do 2~3 sets of 10~12 rep of 20 and 25 on each side. Then i became a little stronger and was able to do more.
For sit ups, you can try to do sit ups on one of those medicine balls, any there are bunch different ways you can do them.
I did so called 8 min ab work out or something that my friend did told me. I would do 30 reps of 4~5 different sit ups. So I would end up doing 120~150 sit ups continuously. Initially I could do 20~25 of each sit ups. Then I could do 30, then 35 reps of each sit up. You will have to look up online to see what kinda ab exercises are there.
I would also do lower back exercise. Gym usually have machine that you can do this on I would do 20 reps, 2~3 sets.
I would do, squats with bar on my shoulder. I could do 2 sets of 15~20 with 35 on each side. Then a set of 10 rep. I was always worried that it would screw up my back. I would also do lateral and diagonal lunges with bar on my shoulder (maybe 10~20 each side, or no weight).
Then I would do shoulder exercise ( I don't know the names). one set with 7.5 lb (if they don't have 7.5lbs, just grab 5lb and 2.5lb). 10~13 of each exercise. Then 10lb
These are generally what I would do. Of course I would start every session with running or biking, and jump roping. Cardio was about 20~30 minutes. And I would try to not to overwork...For example, if I do squats then i wouldn't do lunges etc. If I did a lot of running/ sprinting, then i wouldn't do squats, etc.
This is what I did....Go online, and see what kinda exercises you can do, take it to the gym and try to follow that...Don't start with heavy weights. See how much weight you can handle (if you can do 15 reps, that's good). I would try to increase weight by small increment every week or two weeks.