For training, you can check a larger bag since you might want extra clothes. You'll also be bringing training materials back home, so you'll want room for that.

For your trips, buy a good standard 22 inch bag, and then your flight bag will be your small personal item. Some guys have an extra lunch bag. Or buy that cooler bag that can also double as your flight bag. This will be easiest as you really don't want to check your bags when commuting.