Route 66 starts in Chicago at Jackson Blvd @ Lake Shore Drive. It always started at US Route 41, which was Michigan Avenue from 1926 to 1938. After 1938, Route 41 was moved east to Lake Shore Drive, and US 66 was extended two blocks farther to end there, at the scenic shore of Lake Michigan in Grant Park. This made Route 66 a true shore-to-shore route, from Lake Michigan to the Pacific Ocean in Santa Monica. However, the current begin and end signs are at the 1926 endpoint, even though Route 66 ended at Lake Shore Drive for a much longer time than it ended at Michigan Avenue. Despite the current sign there, never in its history did US Route 66 begin at Adams @ Michigan - by the time Adams was made the 1-way route for westbound US 66 in 1955, it had already been extended out through Grant Park. Lake Shore Drive @ Jackson is also the site of the famous photo in some guidebooks of an IDOT engineer taking down the sign that said "End Route 66" in 1976. Although the state begin and end signs for the historic route are at Michigan Avenue, the City of Chicago periodically hangs Route 66 banners on the street light poles on Jackson in Grant Park, and the Chicago Blues Fest has a "Route 66 stage" located on US 66 on Jackson in Grant Park.
So to start following Route 66 where it began from 1938-1976, turn off of Lake Shore Drive onto Jackson. Head two blocks west, admiring Buckingham Fountain on your left, to Michigan Avenue, and zig-zag one block north on Michigan Avenue (in front of the Art Institute) and head west on Adams Street. From there on, Historic Route 66 is very well marked.
Select your time of day with care, and you can avoid really bad traffic in Chicago. Weekdays 10AM-2PM are usually OK. Saturdays can be bad all day, especially in summer or around the holidays. Sunday isn't usually too bad unless the Bears are playing football at Soldier Field. Traffic info every 10 minutes 24/7 on AM 780.
If Lou Mitchels has closed by the time you get there, consider the landmark Berghoff Restaurant, on Adams between State and Dearborn, which was on Route 66 from 1955-1976, when Adams served the one-way westbound traffic on 66. As you make your way out of Chicago, also consider a food stop at the famous White Fence Farm, just southwest of Welco Corners on the original 1926 (Joliet) alignment of Route 66. Look for motels on US 66 starting about in Countryside, and continuing after US 66 merges into I-55 at the I-294 interchange. Between I-294 and Gardner, some original US 66 motels continue in business at exits off of the 1957 freeway routing of US-66, which became I-55 by simply erecting new signs.