Basically, all you need to do is to keep the composter fueled and filled up with organic matter. After a while it will make

Compost

for you which you can sell at a profit to the bazaar.

Just use whatever fuel is recommended below.

Yes, the composter fully works while offline, while on a different island, etc.. You should consider upgrading the Organic Matter Cap and later the Fuel Cap so you can get more compost overnight.

Usually, you want to find something that has a high amount of organic matter for a good price. So, for example, you may want to buy

ES

Enchanted Seeds

over

SE

Seeds

since it means you will need to buy a lot less items overall.