It can be tedious in finding things to add specific parts to specific pets, because what body part is transferred varies based on both the donor and recipient.
However, adding skills is straightforward. More or less if the recipient doesn't have a skill that the donor has, they'll get it. If the donor has more than one skill that the recipient is missing, the recipient will sometimes get two skills in a single fusion, but other times you need multiple copies of the donor to give them all the missing skills.
This principle is also kind of true for body parts. A humanoid with 13 parts generally won't get new parts from another humanoid. They may get more legs or hands from a horse or asura, who have a second leg slot or four hand slots, respectively (thus parts which the humanoid recipient doesn't have). But still dependent on the above-mentioned variability between donors/recipients.
If you're primarily transferring skills, you'll want to pay attention to any body parts being added to make sure the pet doesn't go over 13 parts total, as they then gain a speed malus. However, this isn't a big deal and if you're just fusing humanoid pets (particularly those bought from derphy) for skills, it is not likely to come up.
Right now I'm just trying to get my pets all the skills I want them to have that they may not have yet. This is detection, pick pocket, fishing, gene engineer (these give them a chance to gain materials when hitting things, E+ only; also not sure if they just need one or if they stack or if they provide different materials), traveling and swimming (speed gains), martial-related skills (eye of mind and tactics if melee/throwing, or marksman if ranged). Cooking. Tailoring (pets may create items for free when you sleep, doesn't appears to be true for the carpentry/jewelcraft though; E+ only). Almost all of these are available on pets from the slave master in derphy. Fishing and swimming can be had from flying frogs. Not sure where to get gene engineer for pets, though.