Great name for a horse - my little black Tennessee Walker is also named "Badger."
You can set up little obstacles for yourself and your horse - stepping (or jumping) over logs, ducking under low branches, walking across streams or up on big flat rocks. Depending on the terrain, it's nice to have some steeper hills to navigate, so you and your horse can develop balance together. Water crossings are great - be aware of the depth and type of footing underwater so you don't find yourself stuck in a mudhole.
I used to clear trail from horseback - using a handsaw to cut smaller low-hanging branches and then drag it while riding my horse until I could find a more open area to drop it out of the way. I don't advise trying that until you know your horse is "okay" with having something dragged beside him in a more contained setting (like an arena/pasture), but it's a useful thing for a horse to know.
Out on the trail is a good place to practice standing tied for a while, even if you're not too far from home. The place where I used to board my horse didn't have good grass pasture, so I would ride my horse out to a nice grassy meadow, dismount and let him graze for a while with just his halter on. Take a book and pack a lunch and just hang out there for a bit.
If you're forging any new trail, just be cautious if you live in an area that used to be farmland, because there may be old coils of rusted fence wire laying around.