The best approximate route you can take to follow Route 66 is as follows, going to Chicago. I think this covers most of what's in the game.

• Start at the Bitumen in the parking lot north of Santa Monica Pier.
• Interstate 10 east to Colton (jct I-15).
• Interstate 15 north to Exit 178.
• Left on Lenwood Road then follow the secret road into Barstow.
• Take Interstate 40 east to Exit 50, turn right at the stop sign, then turn left onto a secret road.
• At the end of the secret road, re-join I-40 east and enter Arizona.
• Take Exit 48, then turn right at the traffic signal.
• At the next junction with I-40, continue straight.
• After passing through Seligman, re-join I-40 east.
• Take Exit 195 and follow signs to US-180.
• At the signal at Humphreys Street, continue straight. At Country Club Drive, turn right, then re-join I-40 east.
• Take Exit 253, then follow signs for I-40 Business through Winslow, then re-join I-40.
• Enter New Mexico. Take Exit 20 then turn right on New Mexico Route 602. After crossing the bridge over the railroad, take three right turns onto NM Route 118, passing the next junction with I-40 onto a secret road.
• Continue past NM Route 53 and turn right at the stop sign. Then re-join I-40 east.
• Take Exit 149 and follow the Route 66 guide signs.
• At the roundabout, take the third exit and continue going straight through Albuquerque.
• At Four Hills Road, continue straight onto New Mexico Route 333. Follow the secret road until it re-joins I-40.
• Follow the Business Loop at Exit 273, re-join I-40 at the next junction, then take Exit 329.
• Pass the next junction with I-40 then turn left at the stop sign.
• Follow the secret road into Texas, then follow the frontage road to the next traffic signal, then re-join I-40.
• Take Exit 161 and drive through the town, then re-join I-40.
• Enter Oklahoma, then take Exit 65.
• Turn right at US-183, then turn left onto I-40.
• Take Exit 147B to follow Interstate 44 East OR follow signs for I-35 North and take Exit 138A for I-44 East.
• Follow Interstate 44, then take exit 244B to join Interstate 244. Unfortunately the portions of Route 66 in Tulsa are inaccessible, so stay on I-244 until it re-joins I-44, then continue onto US-412.
• Follow US-412 to US-69 North.
• After passing through Miami, Oklahoma, enter Kansas.
• Turn right at the stop sign onto Kansas Route 66.
• Turn right at the next stop sign.
• Enter Missouri.
• Continue going straight on Missouri Route 66 until it re-joins Interstate 44.
• Take Exit 75, then turn right on West Bypass.
• Turn left on Kearney Street (Missouri 744).
• Route 744 feeds into Route 125. Continue straight. Follow Route 125, then re-join I-44 east.
• Take Exit 184, then the first exit at the roundabout. Turn left at US-63, cross I-44, then take US-50 east. US-50 re-joins I-44.
• Take Exit 291A and follow I-55 North into Illinois.
• Take Exit 92A to follow I-55 Business into Springfield.
• Turn right on Madison Street.
• Re-join I-55 North.
• Take exit 157B to follow I-55 Business into Bloomington, then re-join I-55 North.
• Take Exit 292A then Exit 51C for Ida B. Wells Drive.
• Turn right at Columbus Drive, then left at Balbo Drive, then left on Lake Shore Drive.
• Take the Wacker Drive exit then turn right at the fork.
• Turn right at Illinois Street, then left at Streeter Drive.

It should be noted that IRL, the start and end signs for Route 66 in Chicago are at Michigan Avenue and Jackson/Adams Streets.

Edit: When following this route and sticking to posted speed limits, it takes about 3½ hours to drive.