You only level weapon skill if you attack with that weapon. If you do 50 damage with an axe, you gain 50 xp for axe skill. If your followers do the damage, you get something like 1/3 of the damage they did added to weapon skills of both of your equiped weapons. Doing the damage your self is fastest way to level, but letting alies do the damage can passively level up even your unused weapons.
Wellness is new so it might be unbalanced. You can eat one food item every week by using it from your inventory. The better the food, the more wellness you get. Meat pie, cabbage soup and milk are probably the best food items right now.
You can build shower as soon as you learn electricity and build the basic electric network. Roman will teach you that. You can exercise if you survive for one year and go north of Riverstone. New training facilities will appear there, raising your atributes and giving you wellness.
Increasing pantry quality means leveling it up. Once it gets above 50 I think, it levels up and you get +5 mood and wellness.