Are you interested in recommendations of existing Pokemon-alike games to see what other people have done to vary the concept? I've played at least 6, maybe more like 10 if I stop and search through my memory.

As far as combat goes, my favorite system for a pet game is tactical turn-based combat, particularly that in the Disgaea series. This isn't primarily a pet game and has no breeding system, but you can create original units and certain monsters that you have defeated in combat become unlocked as unit options. So you can theoretically have an army of all dragons and similar monsters. (This all applies to Disgaea 1, I have #2 on my to-play stack but haven't gotten around to it yet.) If I remember correctly, (and I'm not sure I do...) monsters can't be modified other than by leveling them up and equipping them. Equipment can be improved by adding defeated bosses or other gear to it to increase its stats. So, not a particularly interesting item system, and no skills system aside from one pre-determined sequence of skills per monster type.

Other variants on a turn-based tactical combat system with pets or summons as units: Eternal Eyes (PS1), Monsterseed (PS1), Dofus (MMO) (if playing class Osamodas)

My second favorite combat system for a pet game is where you as the monster go around killing other monsters and eating them. This is arcade/action combat. Tokyo Jungle (PS3) and E.V.O. The Search For Eden (PS1) are the two best examples, though they are quite simple systems with almost no variation in combat from one monster type to another. For a more complex system you'd have to look at a non-pet fantasy action RPG such as Vindictis (MMO), Tera (MMO), or... I'm sure there are good single-player games of this type but I guess I haven't played any. Skylanders, maybe??

As far as visual systems for monster looks go, I strongly favor the kind where the monster is built out of various pieces, then colors and patterns are applied to the shape. For example, a monster might have body shape A, head shape B, foot type C, tail type D, horn type E, and wing type F, plus base color purple, pattern1 stripes, accent color1 green, pattern2 socks, and accent color2 yellow. Spore is the best example of combining body shapes in a 3D system, though it's over-complex for what a pet game really needs. Spore's ancestor-games Creatures (old PC game series) and E.V.O The Search For Eden (PS1) also both feature 2D body part systems. Other games with monster body part systems include Critter Forge, Monster Breeding, and Celebrity Pedigree. (All free online.) 2D color and pattern systems (with genetics and breeding) are quite common among virtual pet sites, e.g. Flight Rising, Tygras, BeastKeeper, and OviPets. (Again, all free online games, as pretty much all virtual pet sites are.) Sadly, not one virtual pet site has a good combat system (IMO), with the exception of ones which use racing/jumping/agility courses instead of combat. Good examples of racing/jumping/agility gameplay (among a ridiculous number of crappy examples...): Petz Horse Club (PC) and Riding Club Championships (Free Online). The chocobo racing minigame in Final Fantasy 7 is also nice, though the difficulty isn't very balanced.

Other pet-related games:

Breeder Tycoon:

Fish Tycoon (PC)

Plant Tycoon (PC)

Eater phone game:

Pocket Frogs

Zen Koi

MMO with humanoid avatar plus multiple automated followers and merge system:

Eudemons (MMO)

MixMaster (MMO)

Azure Dreams (PS1)

MMO with humanoid avatar dps/healer plus one pet tank:

WoW

Perfect World

Single player pet capturing:

Miscrits (free online)

Single player pet merge system:

Monster Rancher series (PS1 and 2)

I'm leaving out the pet grooming/feeding sim and fishtank sim types, as they don't seem to be relevant to what you want to make.