Gathering is simpler now that you can quick gather. Leveling Botanist first is a good idea since it directly compliments Culinarian and doing Botanist levequests is probably going to level you the fastest. Once you can gather the majority of the materials you'll find the time expense to be justified. Players are overcharging on the Ishgard rebuild gathered mats on the MB at the moment because they know players want to unlock the script rewards as fast as possible so its better to just gather your own materials.
Also make sure to pick up the Botanist quests while leveling since they give you equipment you can use and also give XP in addition to the story.