Well, the first thing you need to know is that Route 66 no longer exists--although stretches of it still do--and there are freeways (similar to European motoways or autobahns) that cover most of this trip.
Basically, when you leave Chicago, you need to get on US Interstate 44 (I-44) heading south, then connect in Tulsa, Oklahoma to US Interstate 40 (I-40) westbound until you get to Barstow, California. From Barstow, California head south on US Interstate 15 (I-15) south through the rest of the desert and down the mountain range. When you get past this and reach a town called Cucamonga, you get off on Foothill Boulevard and head west all the way until you get to Pasadena, California. Then you will have travelled most of the same route as Highway 66.
Some of the most scenic spots of this trip are heading through the smaller towns of Arizona and New Mexico--like Winslow, and Flagstaff (both in Arizona), and Gallup (in New Mexico). Also, I suggest you take side trips off this route, like north to Santa Fe, New Mexico and north to the Grand Canyon (when you get to Flagstaff).
Look through the past posts--there was another "thread" where someone asked about Route US-66--and the others on here gave some good suggestions--though I don't remember all the advice now.
Enjoy