I've driven most parts of Route 66 between Joliet, Illinois and Williams, Arizona, some sections in Missouri and Oklahoma dozens of times because it is how I get from here to there, others a few times for the heck of it or as a tour.
A lot of the time you may be on the parallel Interstate because it is simply a better way to travel, sometimes because it is the only road there. Often the old route is the business route through each town bypassed by the Interstate, and in some places the old road serves as the access road.
The places to avoid on 66 are all of the big cities: St Louis, Tulsa, OKC, Amarillo, ABQ, etc. Staying on 66 around around major cities is a real problem because what was countryside 60 years ago is now miles and miles of suburbs with frequent traffic lights, half mile or closer, and in the cities themselves often a light at each city block. Take the Interstate or an expressway through or around.
What there is to see depends on what is interesting to you. Museums? Antique shops? Kitsch? The old towns? Grand Canyon? Meteor Crater? Caverns? What's there will be mostly historical, many attractions that appealed to 1950's and 60's travelers are still there, and the major cities still have most everything they had then and a lot more.
For what to see and do, Michael Walliss's "Route 66" is the classic tour description. For how to find the many different routings of the highway, and the remaining sights along each, I prefer Jerry McClanahan's "EZ66 Guide for Travelers" published by the National Historic Route 66 Federation.
For information on the WWW, try the websites of that Federation, the state Route 66 federations or associations for each of the eight states passed through, or the web sites of the tourist offices for each of those states.
Almost all of the last 1960s alignment in Missouri is still there to follow. Beyond Joplin, you will not find old 66 running alongside I-44, it goes west to Joplin and into Kansas (why TexasShadow lost it at OK-MO line, it didn't go that way).
All 13 miles of 66 through Kansas is still there, pretty easy to find even without a guide, it is well marked.
All of the 60's 66 from the Kansas line to NE side of Tulsa is still there, and you can follow the city street alignments through Tulsa until you have to hop onto I-44 to cross the river, then you can get back off and follow at least one historic alignment of 66 almost all the way to Texas.
In the Texas Panhandle, there are places where the 60's alignment is parallel to I-40, places where it is service road, and places where it is lost. Some of the places where more modern alignments are no longer open to traffic, you can fall back on 1926-1930's alignments, but some sections will be dirt road.