The bike/walking paths along Allen Parkway are a good place to start, and they are accessible from downtown. Just riding around downtown on a weekend morning (much less traffic) is fun. Thereís also some bike paths along Braes bayou Ė you can take them all the way from the UH main campus to the southwest corner of the loop. Memorial park has some singletrack MTB trails and a picnic loop that a lot of people ride. The weather should be great this weekend, hope you can get out and enjoy it.
I mostly ride a road bike Ė that requires getting out of town to get away from some of the traffic and potholes. So Iím not the best source for routes in town.