The Timpanogos Cave Visitor Center is located in American Fork Canyon on UT-92. It sits at the base of the cave trail that leads to Timpanogos Cave, and is closed during the winter season.
See https://www.nps.gov/tica/planyourvisit/hours.htm for the most up-to-date information.
Directions
From I-15: take Exit 284 (Alpine-Highland exit), turn east on UT-92, drive 10 miles (16km) to the monument.
From UT 40: from Heber or Provo Canyon travel on UT40 through Provo Canyon, turning west onto Orem 800 North (UT 52), then after 2.4 miles turn right onto State Street (Hwy 89). After 3.8 miles, turn onto Pleasant Grove 100 East (UT 146). Stay on this road for five miles, as it turns into Canyon Rd and takes you to the mouth of American Fork Canyon. When Canyon Rd ends, turn right into the canyon.
GPS Coordinates
Latitude 0° 0′ 0.0000″ S
Longitude 0° 0′ 0.0000″ W