I use different criteria for every type of creature I tame and the criteria change based on the function I use the pet for.
If the pet is a tank, the absolute most important thing for me is how high the resists are. For example, I have a greater dragon that's 7 off max possible resists. Even though this dragon has only 700's hp, I would choose it over a 900+ hp dragon with 15 off max resists.
If a pet is intended to be a high dmg pet I hit the spreadsheet to figure out the impact strength will have on the pet's damage. For example, the difference between a max str and min str cu is only 1 point dmg per hit on avg. I don't even look at str on cu sidhe... On the other end of the spectrum, I won't tame a frenzied ostard with < 161 str. The reason is that a 161 str frenzie with a 5 pack will hit for 3 more dmg per hit on avg, which means if you have a pack of 5, you added 15 dmg per round on avg. It took me a month to build a pack with decent resists, good hp and high str, but they hit SOOOOO hard. Also, you can bless them to jump to the next dmg threshold.
The last thing I consider when farming a pet is what that pet should specialize in.
White wyrms are excellent for tanking cold and physical damage, so I won't tame a WW that doesn't have near 90 cold and near 70 phys resist. Also, WW have such a small range in every resist that it just makes sense to farm them for one that's near max resists.
Rune beetles are the kings of tanking poison dmg, so you really want to get high poison resist on them. A properly picked rune beetle can tank putrifier.
Bake Kitsune are the kings of fire resist and overall resist cap. I won't tame a bake unless it has at least 55 in every resist and near 90 fire resist. They are an excellent overall tanking pet and can fill in as the tank when you want to use another pet to strike. For Example, if you farm for a strong striking beetle (high str, int, eval, etc) and pair it with a tanking bake, you can really kill some stuff fast.
Well anyway... Pet selection is a very complicated topic. People have written books on less complicated subjects.