Core strength is your center of power. Your core is what stabilizes your trunk giving a solid platform for the moving joints to work against. The best example I can think of to define it is to have you lay down on your back. Now imagine I was going to step on your stomach. You would automatically tighten your core. Abs and lower back are involved, but it is deeper than that.

A strong core in cycling will support more of your weight so less is on your hands.

Do you need to push off the bars to sit up?

There are lots of core exercise routines available online. http://www.livestrong.com/article/18...cise-routines/