There isn't necessarily a best breed for any pet. It depends on the pets moves, the breeds available and the situation. Sometimes the best breed might depend on which moves you are taking.
For example the Blackfuse Bombling. If you are using Explode it only scales with Health so H/H is the best breed and is one of the highest health pets with that ability. On the other hand most everything else does better with a P/P breed so both have their uses and it depends on what you plan to do with it.
There are some general rules, if you will, such as pets with avoidance moves or buffs/debuffs or moves that do X if you go first all do well with Speed. Going first with Dodge or Crouch or Blinding Poison gets you an extra round. There are also moves like tail whip or counter strike that do more damage when you are slower. Undead pets generally do better with Power or Health as their racial kinda makes speed pointless. No benefit to attacking first on your last round if you just rez and attack anyway. The Magic pet racial doesn't pair well with H/H breeds as it is less likely to actually save you any damage. There are more "rules" but none of them are actually hard and fast.
On the topic of Speed it's different in that it has a hard cap so to speak. If you are 1 point faster than your opponent or 50 points faster it doesn't matter. You go first either way. When you know your opponents speed ahead of time, PvE, you can min/max the breed to have as little extra speed as possible. In PVP you don't know who your opponent will be so you may want to be as fast as possible depending on your plan.
The tamers are a good source of stones and can be useful to upgrade pets of the right breed but there are also a number of pets that have to be stone to get a rare because they only come in lower quality versions (link below). I'd plan on using your stones on those first or on exceptionally hard to catch pets like the minifernal or Unborn Valk or maybe something like a Flayer Youngling where it comes in all 9 breeds so trying to get a Rare s/s breed is pretty hard. The flayers move set means s/s is generally preferred. viewtopic.php?f=2&t=2978