When you first start
Digimon World Championship, you'll find that you've been given a few different cages, food, medicine, hunting equipment, and one digimon. The cages you start out with consist of a main holding area that any captured digimon and post-battle will go into, a running track, an infirmary, and 2 more workout areas. Each of the training cages raises and lowers different stats. You'll notice that to raise one stat, you end up lowering another. You'll have to develop your own strategy to getting your digimon to the levels you want to digivolve them.
There is no Story mode in Digimon World Championship. Your main goal is to win the world championship. To do that, you'll need to hunt, train, and battle your digimon. You'll start out with only basic hunting equipment, so don't bother trying to get the higher level digimon with it, because you won't be able to pull them in without breaking the rope. I suggest starting out by capturing the small ones. You do have to keep your capacity in mind, though. At first, you'll only be able to hold about 4 digimon. You can actually hold more, but then when they digivolve, you'll lose that one because you exceed capacity.
After you've captured what you need, you'll need to work on training them. Essentially you'll need to drag and drop them between cages to train. After their stats have gone up/down, you'll need to move them to another cage to change them again. Simply leaving them in a training cage won't continually raise the stats like it would in Digimon Dusk. Make sure to keep an eye on their health too. If it is just low, you can drop them in the infirmary to have it continually raised to max. If they're in need of a bandage, though, make sure to heal them quickly. If you don't, they could die.
After you've trained them a little, you're going to want to battle as much as you can. A digimon has to have been in at least 10 battles to change into a digiegg, so you want to make sure they've been in at least 10 a year. Battles can be one on one, three on three, or one on three. You can battle in tournaments on the days that they are held. Otherwise, you can battle against other trainers. Wireless battles against friends do count towards the battles you need. Once you've trained your digimon enough, you can finally win the world championship.