Looks like your pet is a lab rat
You have 4 real options for training:
Lab Ray - Looks like your pet has been labbed for a year or two. Labbing will bring up your pet's Strength and Hit Points, which are the only important stats for 1-player battlers.
Kitchen Quest - You can do 10 Kitchen Quests per day. Usually you'll get 2-3 stats, sometimes more or less. If you plan on K-Questing, you should only have 1 pet to make sure stats don't go to the wrong pet.
Training School - This is the slowest, but usually cheapest and most reliable method of increasing stats. If you're doing only 1-player, you'll first need to bring up your level (quite a bit!) before you can train anything else. This will take a long time, but it's cheap
Neggs/other stat-increasers - These are the quickest way to increase stats, but are also the most expensive. If you plan on bringing up a lot of stats at once for your pet, you'll need something in the range of 5-10 million (and likely more if you keep going) to spend.
If you're doing 1-player only and the Battledome isn't really important, I recommend just labbing because Defence and Level aren't very important stats in 1-player. If you want to balance out your pet's stats, K-Questing and training would be best (I suggest avoiding the lab, although others may disagree). Negging is only if you have a few million to spare and are really impatient about training.