There are a number of mechanisms at UTSA for gaining research experience. These include, but are not limited to, the UTSA Honors College, College of Sciences Undergraduate Thesis, Volunteering in a Lab, Summer Programs and Year-Round Programs. Also check out the UTSA Office of Undergraduate Research Student Opportunity Center.
Advice on how to approach a faculty member to ask about performing researchDepending on your field, there may also be short term (generally summer) research internships found in town, throughout the country, and even throughout the world.
See an extensive list of summer research opportunities at other locations
NOTE: Most of these pay you, rather than you having to pay them, and some will even pay for a summer course at their institution.
Note on Animal Research and the Biosciences: The animals used in research at UTSA, mostly small vertebrates such as mice and rats, are treated exceptionally well. Some research laboratories at UTSA work with animals and some do not, depending on the questions that are being explored in the lab. Experimental models used at UTSA in the biosciences also include plants, bacteria, fungi, cultured cell lines, and even computer modeling of biochemical pathways. If you have an aptitude for research, don't count yourself out because you don't want to use animals... just choose a field that requires a different experimental model!