You know, to use up excess material.

One idea I had was a potion brewing place, where you could build it, then use materials to make potions.

Another idea I had was some sort of training arena to improve a particular stat, similar to the berries, but that it would also wear your pet out, so it would take 5 energy to train.

One last idea was for a contest hall where you could play different games to earn gold and items. Maybe this one could be part of the starting kit however.

How about a farm which would produce either several regular foods or one stat bonus food a day?

The farm suggestion I favor greatly. I would really enjoy having my own little farm to tend to every day. It could produce a small amount of goods to help supplement the exploring. You could even grow what you wanted, like wheat or sunflowers. Buy seeds from the State Store, or find them exploring. Plant and in a reasonable time, reap the rewards of your harvest. I'm not sure about anyone else, but I tend to spend a bit of money to feed my creatures. This will continue as I get more creatures and you're not guaranteed to find food while exploring all the time.