Digimon Masters Online

Wall o' guide:
Spoiler Alert! Click to view.
Partner Digimon:
Your digimon will gain access to their digivolved forms at set levels.
Champion at 11, Ultimate at 25, Mega at 41.
Some Digimon have a Side-Mega form, which is another Mega form that tends to have better stats than the first. Others have a "Burst Mode" form (such as your starter), which will have better stats and more powerful skills than their Mega form.
To get access to Side-Mega or Burst Mode you'll need to complete a quest or have a certain item depending on the Digimon. Items are cash shop only, but can be bought in game from other players. Your Digimon also needs to be 55/65/71 to gain access to this form, depending on the Digimon.

Your Digimon will also gain skill points as they level. Use these to increase the power of the Digimon's skills by opening your Digimon page by pressing D and checking the skills tab.
Not all Digimon will have all their skills immediately; some will learn new skills as they level up. Most Digimon will only have 2 skills total.

Enemy Digimon:
Normal enemy Digimon won't aggro.
Digimon with (Leader) after their name will aggro.
Digimon with anything else after their name tend to be involved in quests or summoned from monster cards.
An enemy Digimon's name is a quick way to get a rough idea of their level.
If the name is white, it's roughly the same level as you.
Light blue, means it's a slightly lower leve.
Dark blue means it's an even lower level again.
Grey means it's so low it won't give EXP or drop items (except quest items, which will always drop regardless of level).
Light red means it's a slightly higher level.
Dark red means it's an even higher level again.

DigiEggs:
When you defeat enemy Digimon, they can drop DigiEggs.
Cracked DigiEggs have no use, other than to be traded in for money. Class 1 will give 25 bits, class 2 will give 100, class 3 will give 150. Haven't seen any higher yet, so can't give any info on those. ;p
Next, you can find (family) DigiEggs (Dragon DigiEgg, Beast DigiEgg, etc). These can be returned for money, or scanned for items (which I recommend doing).
These come in different types, as indicated on their description window.
DATA will scan into DATA of that family. DATA is used to hatch Mercenary DigiEggs.
Low Class will scan into common items, such as food or low power healing items.
High class will scan into rarer items, such as DATA, healing items, and rarely clothes.
Super Class is the rarest type of item DigiEgg and will scan into clothes, DATA or chips for your Digivice.

The last type of DigiEgg you will find (and arguably the rarest of the lot) is:

Mercenary DigiEggs:
These are the eggs you use to hatch extra partner Digimon. You can get some from quests, buy some from shops, or find them as rare drops from enemies.
To hatch them, you need DATA of the type and quantity (x3 at least) listed on the egg.
When you have enough DATA, you go to an Incubator at the DATS Center or one of the villages. You talk to the Incubator and drag the DigiEgg you want to hatch onto it.
Next you'll need some luck. Click "Input DATA" and the Incubator will start to feed the DATA into your DigiEgg. When you input DATA, there's a 50/50 chance of success. If you fail, there's a chance the egg will break and you'll have to find another and start again.
You need to successfully input DATA at least 3 times to be able to hatch the DigiEgg.
If you're feeling lucky, you can try to get 4/5 or even 5/5 DATA inputs done. Doing so will make your Digimon bigger and give it better stats, but obviously you're risking more potential breaks.

Now the most important piece of information ever:
Remember to have fun. :3