The fastest way is using Grand Company Supply and Provisioning Missions. Go to your Grand Company HQ, talk to to Grand Company Personnel and take the Supply and Provisioning missions. They ask for specific items for each crafting and gathering job. Craft and gather those items and deliver them. They reset daily.

You can also crafting and gathering leves for XP.

You can always do it the old fashion way which is to just grind them. For mining and botanist, this is free other than repair cost of your tools. just go to mining and botanist nodes about 5 levels higher than your level and just gather.