Houston is close to the Gulf of Mexico and can get very humid year-round. The summers are hot, but the winter is very mild. Come during spring for some of the best weather. Houston can be very rainy, but it usually comes down as brief downpours, not as soaking rains or drizzle.
Start by joining what seems like everyone in Houston at Memorial Park for a 3-mile jog on the crushed stone trail - shaded by piney woods, the Memorial Park running trail is even lighted. The only problem will be parking, so get up early.
Head over to Terry Hershey Park to use the 10 mile paved hike and bike trail for bike riding. The asphalt path stretches along Buffalo Bayou, and is the best uninterrupted bike riding in the Houston area - Houston is choked with traffic until you get far outside the city. The hike and bike trail is a shared trail though, so look out for kids, walkers, and dogs.
Armand Bayou Nature Center is the opposite of Houston's concrete jungle. Before Houston became the nation's fourth largest city, the area along Galveston Bay was a mix of wetlands, prairie, and piney woods forest. Go for a walk along the trails at the bayou for a glimpse of how it must have looked.
Huntsville State Park, one hour north of Houston, has great hiking, trail running, and mountain biking in the piney woods, surrounding a lake. The best trail in Huntsville State Park is the singletrack Chinquapin trail - the CCC trail is similar in length but has a lot of open jeep/dirt road. The lake is very inviting to swim in, but like all of East Texas, watch out for alligators!
Brazos Bend State Park is another world altogether. A low-lying area along the Brazos River, Brazos Bend State Park is a guaranteed place to see alligators. The trails in the park are great for leisurely mountain biking - nothing too technical makes them great for families (except for the gators).
Unfortunately, the whole Gulf Coast really doesn't have any hills. You'll have to head west 600 miles to Guadalupe Mountains National Park or Big Bend National Park or Davis Mountains State Park for real mountains.
About an hour away, Huntsville State Park has a few rolling hills, but nothing major and certainly no scenic views or anything. You'll have to head west to the Texas Hill Country, or at least Bastrop State Park.
The Galveston beaches near the city are not exactly the greatest beach destination. The gulf water near Galveston can be very muddy, and the sand is usually a dingy brown. Don't go all the way down there for a relaxing day at the beach - many visitors thinking of the crystal clear water and white sand beaches of the Florida Gulf coast end up disappointed. Instead head about ten miles west to Galveston Island State Park, where the beach is wider, and there are showers and restrooms.
The Rice University running trail is narrow and can get very crowded.