Cathedral has parking lots on Lawrence Avenue & South 5th Street. If you park on Lawrence, you will cross the street and enter through the main doors on South 6th Street. The parking lot on South 5th Street is directly adjacent to the Cathedral and you will enter through the side Atrium.
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Springfield does have an airport and rental car options, but it only offers flights through DFW, ORD, PGD, and AZA.
Other airports in central Illinois that might have better flight options include Bloomington (BMI) and Peoria (PIA). Both are about an hour's drive from Springfield and have rental car options at the airport. There is an Amtrak route between Bloomington and Springfield, and a charter bus that runs from Peoria to Bloomington. St. Louis Mid-America in Belleville (BLV) is about an hour and a half south of Springfield.
Chicago and St. Louis have larger airports with more flight options. Chicago (ORD and MDW) is about a 3-hour drive from Springfield, and the drive to St. Louis (STL) is under 2 hours. All airports have rental car options and both cities have Amtrak routes to Springfield.