I forgot to mention it, but start stretching. Especially splits (left, right, middle) and bridges. Splits to help your jumps/flying (maybe) and bridges because a minimum flexibility is required for a handspring/walkover.
40-50 yard sprints are great, because they're directly correlated with jump height. Broad jumps help to build explosive full body strength, and even has a swing associated just like a handspring. Squat jumps are good, as well as calf raises, pliobox jumps, explosive skips, bounds, pretty much you want to build explosive power through your legs, so don't walk/ jog on a treadmill for 30 minutes. It'll help your endurance, but not your tumbling.
Everyone hates them, but push ups (no "girl" push ups please unless that's all you can do) are great for your shoulders and chest. Hand stands are also fantastic for building the muscles that are involved in the middle part of a cartwheel/roundoff/handspring.
Anything you can do for abs, planks, sit ups, reverse sit-ups etc. will help build up your core.
You don't have to workout forever. You can get a workout in in under 20 minutes a day if you do it right.
Say you do five forty yard sprints, two times a week (I usually sprint the forty, and walk the forty back then go again). That should take you around 10 to 15 minutes to warm up ( jog half a mile and stretch) and 10 minutes to do the sprints (likely less). The next week, you should shoot for six sprints. You do 3 sets of 15 pushups( five minutes tops), the next week you should try for 3 sets of 20, or 4 sets of 15.
Just always be improving yourself and you'll be glad of it when you start tumbling.