A strong core is essential to allow your body to carry more weight. You can train cores by plank variations, planche, human flag, etc. If you're looking to compound movements to work on not only the core but other muscle groups, squats, deadlifts, bench press, pull ups, etc are useful as they target different muscle groups to make your workout more efficient and effective.
An important part of the core that's neglected is TVAs. You can train them by doing stomach vacuums. And working on abs, always start with lower abs then obliques then upper abs. The reason to this is to use more energy on the weaker lower abs first. The best exercise I recommend for abs is hanging leg raises, aka hanging on a bar, curl your body as you exhale and bring your knees to your chest and your feet towards the bar.
Sorry for the mass load of info, I'm quite into fitness so....
