The Fiesta at Gessner/Braeswood is a good starting place. The Brays Bayou trail is paved to Calhoun at MacGregor park, but you'll have to change sides every now & then. There & back to your car is a good ride. Later you would explore Hermann Park, Med Center & Rice U and north into downtown.
The Heights Trail continuing to TC Jester is best explored on a separate trip, park downtown and start under I-10 at UH Downtown. You have to get a little creative at W 7th & Shepherd to get to The White Oak trail at W 11th, but it's doable. Keep going west at Shepherd through the grass and follow the bayou, turn right into the townhouse parking area and you'll go north on Worthshire to reach W 11th. From there it's paved until the end at Antoine. Turn around there. Oh - I forgot. The trail is closed at 610 but the roads are fine or go overland for a while.
Memorial Park - park along N Picnic Lane by the softball fields. You might see the S Picnic Lane loop (which is roadie training grounds on weekday evenings) but stay closer to Memorial Drive. You'll have a path from there east to downtown, except for the unfortunate 1 mile sidewalk portion from Crestwood to Shepherd. Watch out at the Starbucks near Detering. There will be other bikes & joggers. Most everyone knows that it's a crowded spot and watches out for others, so you'll be OK, just watch out.