The skill system helps each character be a bit more individual. You can make your character a fist fighting champion, a lock picking master or the best Angler around.
You gain XP for each skill the more you use that skill. For example the more you run the more XP you get towards the next stamina skill.
Do /profile to check your level and progress towards the next level.
The fishing skill is leveled up the more you fish, each level decreases your chances of catching nothing and unlocks different types of catchable fish. You require bread to fish that you can make whilst cooking or purchase from the shops.
Do /togglefish to see some of the fishing locations.
Farming is simple and easy, all you need is some water and your seed. Use the seed, add some water and you're done! Each level of your farming skill helps you understand how to use other seeds as when you first start you will only be able to grow Tomatoes from Tomato seeds.
These are leveled up passively the more you drive and fly. The higher the level the more control your character will have of the vehicle.
(These bonuses are small but do stack up)
Cooking is rather simple, all you need to do is find a grill or oven that you're able to cook at and it will give you some recipes. The higher your cooking level the more recipes you will unlock. Cooking goes hand in hand with Farming as you can use all the stuff you farm to make food. You can also buy the ingredients from the Marketplace.
Very simple, the higher your stamina the longer you can sprint for. You can let this level up passively as you run around or you can go down to the gym and work out for an extra boost.
Strength increases how much damage you do in unarmed combat (fist fights). If you want to build up your strength you can also do this at the gym along with stamina.
Stealth is a weird one, the higher the stealth skill the less likely NPCs are likely to call the police (whilst stealthed). You level this up by simply stealthing about (hit L CTRL to enter stealth)
Swimming is like stamina but instead it increases how long you can stay underwater for. Level this up by swimming (who would have thought)
Thieving helps with anything to do with the lockpick item, for example lock picking cars. The higher the level the higher chance of success you have. Level this up by simply using lockpicks more (they can be crafted a crafting table like the one in your motel room)
In basics the more you shoot the higher your recoil level the less recoil you have
Level this by cutting wood at the Paleto Bay sawmill. It's a great way to make some money.