At certain times in the game, your Digimon will be able to Digivolve naturally. At the begining they can't Digivolve at all. After you beat the mid boss on level 1, you'll be able to reach the Champion level, but at the begining of level 2 Digivolution will be restricted again, until you beat the level 2 mid boss. When you get to the level 2 boss, you'll be able to Digivolve to the Ultimate level. Getting the Mega forms is more complicated, and is covered in the Bosses section. You can ArmorDigivolve at any time as long as you have the right Digiegg.

Now I have pictures of most of the Digimon! I wanted have their LCD pics, but those have to be drawn with a mouse which takes a great deal of time and effort.  So, instead, I have some other pics that came from here. (I don't know what the name of the page is, it's in Japanese ;) )
 
 

Gatomon w/o tail ring (Rookie)  --> HP: 3 AP: level 2           |          \/	Nefertimon (Digiegg of Light) HP: 7 AP: level 3
Gatomon w/ tail ring (Champion) HP: 7 AP: level 3           |          \/
Angewomon (Ultimate) HP: 11 AP: level 4          |         \/
Magnadramon (Mega) HP: 14 AP: level 5

Wormmon (Rookie) HP: 3 AP: level 1         |        \/	(Wormmon has no Armor evolutions)
Stingmon (Champion) HP: 8 AP: level 3        |       \/
Dinobeemon (Ultimate, Jogress evolution with Exveemon) HP: 11 AP: level 4       |      \/
Grankuwagamon (Mega) HP: 14 AP: level 5