Okay so I understand this would be a big change but I think most people would like it and it would add lots of gameplay and replay value to the game in general. It would also give something for the social teams to do.
You would level up these skills by simply partaking in them and there can be like another tool button to track them.
For example,
Fishing skill can make it easier to find rare pokemon, or a faster catch rate, or faster hook time.
Poke'Ball Crafting. It can be a way to make poke'balls to use or sell and the higher your level is the more types of balls you can craft.
Stealth skill can act as a repel activated by choice at will.
Breeding skill can make prices for breeding cheaper or reduce the waiting time for eggs to hatch. It could be increased just by breeding poke'mon.
Foraging skill can make it easier to find seeds and certain rare items. It might also make it more likely to find Poke'mon with better IV's in the wild. Leveled up by picking acorns and similar things.
Item Crafting can be a skill where you make healing items or forge battle items that are held.
These items crafting stuff can be split into other skills too, like a mining skill, such as in Diamond and Pearl.
Cooking skill can be used to create poffins.
Berry Farming can be turned into a skill.
Lure skill can be raised by catching poke'mon only attainable by lure.
Perhaps these skills can have some cosmetic reward when you max them out, like a cape or something with a symbol on it for that skill.
Stuff like that. Yeah.
If anyone has any other ideas they can add it.