I know there is a stat cap - so what is your pet's state spread? Which stats do you focus on and why?

Any recommendations?

It depends on battle style, weapons and personal preference. Most people are in favor of a high health build because healing is based off of the % of your total health where as defense and strength are on a square root curve. In general, the higher your strength the harder you hit so battles can end quicker. The higher your defense the more you block so your enemy does less damage and you could survive long battles. If you have higher speed than your opponent they can't crit hit you and you can crit hit them, causing more damage. Certain weapons also work differently depending on your stats. There are weapons that work better with high speed and other weapons that work better with def > strength or strength > def.

I am doing multiple builds with my pet as some builds are better than other builds for some challengers. Once you are close to cap, you could beat all challengers but it may be more difficult. For example, if you have a high speed build, opponents like prodigy and the mage can't crit hit you and do double damage. However, if your other stats are low then opponents like meatslab and the mutant scientist can do a lot of damage to you.

There is also controversy about how useful speed is.

There are several people that argue that its useless.

The arguments basically oil down to: Speed never helps you win battles earlier and you will be able to win battles earlier than you otherwise would by ignoring speed. But having access to crit hits for global boss battles might be beneficial if you train to 7001 speed.

Haha maybe speed was a bad stat to give an example of since there is a lot of controversy around it and how it works (ie: if your defense is greater than your opponent's strength, is it possible for them to crit hit you if they have much higher speed?) but I think it goes without saying that builds that favor higher health allow you to beat some opponents earlier on. You will eventually beat them all if you keep training but some opponents can be more difficult than others early on depending on what your build looks like.