1. Don't over think it
As many times as I have made kombucha I have had several fantastic batches and a few flops. Sticking with your basics and trusted black tea, sugar, and boiling water is always best for your first few batches.
2. The double fermentation process
The second fermentation is key to getting a bubbly drink. With a single fermentation I find the kombucha often is flat and can easily become more vinegary in flavor. Once you start getting to a decent acidity, removing your scoby and adding your flavoring ingredients for a second fermentation can bring a whole new life to the batch.
3. Don't get discoraged
Jars break, some batches mold, and some just don't turn out. When you start experimenting with flavors make sure you have a bake up scoby and some starter kicking around. I can almost guarentee, even with a solid first few batches you will have at least one flop.
Okay now on to the good stuff.... My top 3 favorite at home flavors!
Strawberry is a consistent flavor and the sugars in it make it great for a second fermentation. My method to making strawberry kombucha is add a 1/2 cup of frozen strawberry slices to the jar and leave for a couple extra days, once your scoby is removed of course. I have also had really good luck with strawberry mint, where you add two stalks of fresh mint to the brew as well.
Ginger is always a safe flavor. It doesn't respond to the second fermentation the same as the strawberry, which really makes a nice effervescent batch. This flavor is more like a ginger-ale or ginger-beer though. I slice rounds of ginger root and at them directly to my batch similar to the strawberry slices, then I wait.
If you find the batch just isn't quite sweet enough adding honey can add more flavor all on it's own. Keep in mind a little bit of raw honey will go a long way so make sure to taste the batches in between.
I have tried other fruits which work very well, careful adding citrus because the acidity in the fruit can alter the pH balance of the brew. Most flavors turn out well but some could use a little refinement as you co along. Herbs also add really nice subtle flavor, adding basil can add a nice twist to some of the sweet fruity flavors.
Other flavors that I have tasted, but haven't made myself include lavender (which I am hoping to try ASAP), pineapple, and apple cinnamon.