I've ridden Rt 66 from St. Louis to Oatman, AZ in one stretch, and from StL to Chicago and back on another trip.
The oldest, most intact stretches are actually in Oklahoma, but being a big fan of the southwest territories my favorite stretches were those found in NM and AZ.
Through IL and MO you really eat up a LOT of time wayfinding and hopping back and forth on service roads along I-55 & I-44. The Urban Sprawl is BRUTAL.
Look for the series of fold-out maps called "HERE IT IS" (I thnk). It gives turn-by-turn directions, along with locations of sights and landmarks along the way, and in a fun format. I took those directions and manually wrote down all the turns and stuff into a word document and pasted together a scroll-map out of it. It sorta worked.
As for time of year - I believe it is the best kept secret of the southwest that the 1st couple weeks of October are absolutely HEAVENLY to be there. Much before that, and it can be brutally hot, after that and you run the risk of hitting a snow storm or 2 and road closures up on the Colorado Plateu.
Hotels and campgrounds are extremely plentiful all along the Old Road, and you won't have to worry about finding something as you ride along.
Sag-wagons are for Hardley-Ablesons. Show some ballz man!
But really, there's more than enough dealers along the way to lend support at reasonable intervals as well, so even if you run into trouble you'll either be able to make it to the next one, or at least get to within the AAA+ 100 mile range.
just do it! You won't regret, nor ever forget it!