Gardening can be leveled up a variety of ways.
A good start is to read gardening books till you're about level 4. but you can also set up a row of bonsai trees and keep cutting them, or have a set of planters in your yard where you grow some plants.
At level 4 you unlock the ability to talk to plants, an interaction that fills both your social need and your gardening skill. You can either spend you day doing that or simply continuing reading and tending plants.
The biggest help I found was at level 8. Once you reach this level, you can regrow bonsai trees instantly for $50. Simply cutting and regrowing over and over seems to grow your skill a might faster.