How do i get pet skills above level 4?

Answer from: HFR1B8
You can open painted egg at Mystery Store. though it's random but you also can get pet lvl 7. I got my Thunder Dragon :D

Answer from: Haizel
When you do a skill synthesis between pets, it randomly assigns skills from each parent pet to the child pet. The usefulness of this is having two or more elements on your pet. Every level 4 pet has at least one skill of level 4. If it's a flame polly, it'll have Fire ATK Lv4... so when you face groups of water or earth based enemies, your attacks will be halved in effectiveness. If that flame polly is synthed with a bubble polly, it has a chance to have Fire ATK Lv4 AND Water ATK Lv4. When your flame polly does battle with an earth enemy, instead of using fire, it'll use the water attack and do 150% damage instead.

The only way to get a higher level skill is to open the egg of a higher level pet, or to synth the higher level pet with your lower level pet, which would be totally inadvisable. At the higher levels of pets, once you have level 7's or 8's in both of your pet slots, if you get another level 7 or 8 egg, you can synth skills so your pets can do big damage to a bigger variety of pets. Having Fire and Water on the same pet means you can do 150% damage to Grass, Elec, Fire, and Earth. Just keep in mind, unless you use Magic Hammers to make sure you get the skill you definitely want on each pet, it's a roll of the dice, and you might not get the skills you're really looking for.