Chicago has train lines from both airports. If you land at O'Hare, you can take the Blue line to downtown. If you land at Midway, you can take the Orange line. Uber and Taxis are also available.
Chicago has a solid public transportation system. The CTA train lines go from both airports to downtown and the Red and Brown lines can take you from downtown to most of the north side of the city. Buses and trains stations are tap to pay with your credit card. You can also purchase a Ventra pass for unlimited rides for 1, 3, or 7 days. More details on fares can be found here: https://www.transitchicago.com/fares/
Saint Clement Church is in Lincoln Park. There is a church parking lot at Demming and Orchard across from the main entrance of the church. If you are coming from downtown, you can take the Brown line or Red line and get off at the Fullerton stop. It is a 12 minute walk to the church. Uber and taxis rides will also be easy from downtown to Lincoln Park.
Saint Clement and Salvatore's are very close. It is approximately a 6 minute walk.
Yes, there will be valet parking at Salvatore's.