Does it have anything to do with their color or job?

Nope, just stats. Strength, Defense, Health, Speed, Intelligence.

Nope. Pet rank is based on their stats, and the only guaranteed way to raise them is to train them, but you can also get extra gains from scratchcards, random events, and Oracle quests (which is a different random event).

You can also raise your pet's intelligence by reading them books or playing mystic spindles and buying int points. Collecting from jobs also has a chance for a random stat. Training is the main thing though. xD

Don't forget battle quests! Doing battle quests every day, alongside training with Jim, is the best way to gain stats. Reischen, in the sacred lands, will give you a challenger to beat, and when you do, you'll get experience you can spend on your pet's stats in the battle coliseum. He starts you with 5 easy challengers, and then it gets harder from there. You won't be able to beat many at first, but as your pet grows, you'll get further and further, and get more and more exp each day, meaning more stats.

Thank you both so much for the help! ^^

I have no idea what you said, but thank you! Cx I don't know my way around that well yet I don't think.

Archer is talking about daily battle quests, which you can access here. You need to make sure your pet isn't at 0 hunger or happiness and has some weapons equipped (in the armory) first. Check out these links if you're interested in battling. c: link link