Sturgis is primarily served by Rapid City Regional Airport (airport code RAP) just east of Rapid City. Travel time from Rapid City Regional Airport to Sturgis is under an hour by car. The airport in Rapid City is a regional airport, and as such airport transportation to Sturgis needs will necessarily have at least one connection to and from most other locations. Rapid City Regional has direct flights between Minneapolis, MN, Salt Lake City, UT, Denver, CO, Las Vegas, NV, Phoenix, AZ, Houston and Dallas, TX, and Atlanta, GA as of early 2016.
To reach Sturgis from Rapid City Regional Airport simply head west on Interstate 90 to exit 32 (first exit), or exit 30 (second exit). To reach I-90 leave the airport via Highway 44, travel just over 5 miles to Elk Vale Road. Turn right/head north on Elk Vale, you will reach I-90 in just under 3 miles.
Sturgis is also served by two smaller airports. The Sturgis Municipal Airport is a small airport approximately 4 miles east of Sturgis owned by the City of Sturgis. Slightly further away, Clyde Ice Field / Black Hills Airport is a slightly larger airport operated by the Lawrence County Airport Board. The Black Hills Airport is just east of Spearfish, SD, about 15 minutes travel time from Sturgis.
If flying in you should plan to rent a car for the duration of your stay. The regional public transportation infrastructure is generally inadequate to serve even the most basic travel needs for a visitor. Those flying into Rapid City and staying in Rapid City may have their needs met by shuttle and taxi services, but the options are more limited once outside of Rapid City. If you plan to visit any of the famous regional sites, such as Mount Rushmore, Crazy Horse, Devils Tower, or Bear Butte, you are almost certain to need your own vehicle to get it done.
Airport transportation to Sturgis is perhaps slightly less convenient than traveling to Sturgis by car for those in the region, but Rapid City Regional Airport service makes air travel into the area quite reasonable. Just remember to rent a car!