The Suwanee and especially the Santa Fe are flat out awesome rivers. So much history and a tremendous amount of wildlife and springs dotting the shoreline every few miles where you can take a break from the heat - including Ginnie Springs which is a pretty famous dive spot. You can start on the Santa Fe River all the way up at the US441 bridge near the town of High Springs and canoe all the way to gulf coast. The crystal clear Ichetucknee flows into the Santa Fe along the way and the Santa Fe keeps on flowing down into the Suwanee River. The Suwanee then flows into the gulf just north of Cedar Key.
You can make shuttle arrangements with the Canoe Outpost in High Springs. They are right on the river at US441. You can park your car there and the owner lives there at the dock so it's a pretty secure trailhead for your car. The outfitter will pick you up in the town of Suwanee on the coast and bring you back to High Springs if you want to do the whole river. It's about 60 highway miles from the town of Suwanee to High Springs. The river miles are of course longer because it winds it way to the coast unlike the highways. He doesn't list a price for shuttle services all the way from Suwanee but talk to him and work out a price.
If you do this trip make sure you stay at Manatee Springs State Park for a night. Awesome campground - especially in winter when the manatees hang out at the spring. If you have a chance, go for half day of tubing down the Ichetucknee. You won't be able to paddle up the Ichetucknee - it is too fast. But you can stash your canoe in the woods and walk about 2 miles to the state park and rent a tube and float back to your canoe. Ichetucknee has different parts of the river open depending on time of year. In Winter they close the northern half - which is the really pretty part of the river.
Also, if you do the entire trip - High Springs to the coast - Do a very very very slow float if you can. There is so much to see and so many springs to stop at - you don't want to rush it. You could easily make it into a 14 day float trip and never be bored.
By the way, if you do this trip in winter, you'll definitely need a wetsuit. The springs are cold year round - 72F water. You may want a wetsuit year round but for sure you'll need one in winter no matter how hot the air temp is.
hope that helps