There are opportunities for hiking and biking all over the San Antonio area. Here is a link to the Howard W. Peak Greenway Trail System that contains maps and information on trail safety.
According to the site, "Forty-one miles of developed greenway trails are now open to the public. The Linear Park Development Program for Salado Creek, Leon Creek, Medina River and the San Antonio River is funded through sales tax approved by voters in 2000, 2005 and 2010. Approximately 1,200 acres of property along San Antonio Creekways have been acquired for the program since its inception."