As a variant rule, you will have to spend downtime training or studying before you gain the benefits of a new level.
The training time and cost depends upon the level to be gained (DMG 131)
This represents you seeking out trainers, doing independent study to learn new tricks of the trade, repeatedly lifting heavy things and putting them back down, communing with the spirits, and such.
If you want, look at the abilities you are about to gain and work with me to come up with a little story of what sort of training you are doing. This can be an opportunity to meet new helpful NPCs, find new sources of info, learn more about how your class and character fit in the world, and more. So you might seek out a bookstore with tomes on magic, attempt to contact local druids, participate in a mounted chocobo race, or have a dark vision. This sort of training roleplay is a great use of Downtime between sessions.