I pick my times on the greenways to minimize pedestrian traffic, know where to expect it, know all the blind curves, climbs, drops, bridges. The crowded trails on the weekends I completely avoid. Always happy to see a confident, well-trained heel hound (mine always were). One nice thing, is many dog-training classes meet at the trailhead parks - those dogs and owners are wonderful, and a good example to the lame. Aside from being the bane of greenways, leashes longer than 6' aren't legal, but that doesn't stop the mindless from creating hazards. Cycling groups larger than 2 or 3 people are also dangerous, because they pass pedestrians as a mindless herd, on bridges, etc.
San Antonio is doing a really good job on our greenways, and eventually we will have 150 miles connected, making it easy to choose good remote rides.
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We have one trail that is a national park, connecting the 5 missions, though don't make the mistake of riding this mid-day in the summer, because there is like only two trees remaining along the lower San Antonio River - the channel was scoured long ago for flood drainage from downtown - though they have built some nice gazebos along the trail. The trailhead is also at one of the better brewpubs in San Antonio, Blue Star, and there's a nice bike shop there, as well.
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https://www.nps.gov/saan/planyourvisit/hikebike.htm
http://www.sanantonioriver.org/images/SARIP-map/SARIP Full Map-web1.pdf
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Guess that brings up another tip - know the weather and plan for it. Clouds, sun, wind, rain, temperature swings. Look ahead hour-by-hour on the weather channel website or somewhere to know what to expect in the ride your're planning. Sailing 10 miles downwind, then turning around can be a shock, and a test of your late-ride endurance (especially for families with young riders).