
What is the most important thing to work on physically
How do you improve core strength? Push ups?

One solution to this. Olympic weightlifting, It addresses all these areas.
This I noticed after I started training
-Flexibility (less injurys)
-Low post strength.
-Legs did not get tired anywhere near as fast as they used to
-Increased vertical leap. Added about 5" of vert.
Dont want to Oly lit? Squats, deads and power cleans.
Plyometrics would have helped more again but only so many hours in the day.

Definitely core body strength and explosiveness in upper and lower body. Power training which is strength and quickness would be best for basketball. Flexibility exercises help this also.

I'm also guessing that learning to catch the ball is also something to be worked on or you'll never get the ball or the trust from your team mates

All those you mentioned go hand in hand, as Anon said you can't isolate them.
Without "flexibility" you won't get "posture" and without those two you won't get endurance.
At the same time you need flexibility to achieve core body strength otherwise soft tissue injuries will be an every day headache.
Don't concentrate on any specifically, depending on the age let the kid grow up naturally, make sure he/she does warm up before stretching, by doing that flexibility will follow.

For kids, definitley core body strength.
