Everybody has said basically everything you needed to hear. Bodyweight exercises are as absolutely essential as Ironman said and it shouldn't be avoided. Yeah, the core as well needs a lot of work. Good thing about the core is that it can be worked every other day, unlike legs, back and chest.
You want to build your FUNCTIONAL strength, though. Fedor doesn't look all imposing, but he is so strong and has a lot endurance in his strenght. A lot of that is down to simply wood choppin' to work the core and lots of dips and pull ups.
Squats are a must, but from the research I have done, Front Squats is the best in terms of a weight lifting exercise carrying over for atheletic performances, due to the fact that there is great emphasis on your. GSP does a lot of Front Squats and has claimed that it's the exercise most responsible for making him strong.
Why Do We Fall?