I've encountered several different pools in my play through. Is there a difference between the pools, say "secret" pools vs "promising" pools?
Different fishing pools provide different grades of rewards, both fish and items, when fished.
The lowest grade can be caught in the regular pool found in towns. Promising pools provide better rewards, while secret pools provide the best (currently known) rewards. If you do find another kind of pool not listed here, you can safely assume it's even better than a secret pool!
Regularly, you will catch normal-sized fish, which transform your pet for 5 minutes. Better pools have a higher chance of rewarding Big fish, which transform your pet for 15 minutes instead of 5, or Giant fish, which permanently transform your pet. Also, you have a higher chance of catching rare and unique items in better pools. Secret pools tend to produce many more items than fish, although they have the best chance of catching Big and Giant fish.
Another question already provides a list of fish you can catch, but I added it here for completeness.
The most common kinds of fish that you can catch in normal, Big, and Giant sizes are listed below, including what creature they transform your pet into:
There are also some special fish that you will catch less frequently: