If you run or bike, take a left as you leave the driveway and then an immediate right and another right on to Ranch Road 1, which turns in to Park Road 1 and takes you through the LBJ Ranch right along the Pedernales River. You're likely to see deer in the early morning hours. The road has very little traffic, which is why I run it myself.
Enchanted Rock provides a beautiful view of the Hill Country, but arrive early on the weekends or you might not get entrance to the park - it's pretty crowded. Check the Enchanted Rock Facebook page for special events such as stargazing and moonlit hikes. When the weather is warm, take a dip in the swimming hole at Blanco State Park.
Old Tunnel State Park is off Old San Antonio Road just east of Fredericksburg. The seasonal (May - November) emergence of bats from the tunnel at dusk is worth the visit. There are plenty of state parks nearby where you can go fishing, canoeing, swimming, hiking and more.
Stop in at Wildseed Farms to enjoy the best in seasonal blooms. Pick up your own seeds to plant in your backyard...you can have your very own field of bluebonnets.
If you're visiting during bluebonnet season you definitely need to drive the Willow City Loop. Beautiful Hill Country vistas, longhorns, boots and bluebonnets. Stop in at Harry's on the Loop for a cold beer before you head back.