What are some good drills you can do in order to ...

Here is an easy one:

Place a chair in front of you (for safety reasons). Close your eyes and try to stand on one leg for a few seconds (about 10).  

Change legs repeat 3-4 times on each leg.  

Use some weights (aprox. 1 kilo) and stand  on one  leg while you alternate punches in the air above your head. Then you can also switch and give punches forward. Try it 10 times on each leg and finally you can punch sideways as well.

You can read this interesting article for more drills:

http://www.pponline.co.uk/encyc/balance-exercise.html 

yoga!

lean forward while standing on one leg.. then switch...

(as someone who has a terrible core I Use a Bosu ball and stability ball a lot in my excercise routines while lifting weights to help balance and "engage" the core muscles). Barbell curls - stand on one leg and do a set of 10, switch legs and do another set of 10. Practice safety first...