I have a Scotts 20" Classic Reel Mower. The blades are starting to dull a bit.
What's the best way to sharpen the blades on the reel?
Adjust the cutter bar
Often times bad cutting results are caused by a mower in need of adjustment, not a mower in need of sharpening. The first thing you'll want to try, is to adjust the cutter bar to ensure you have proper contact with the reel. This is often done using a few adjustment screws, and a tape measure or spacing tool (that may have come with the mower).
If this does not provided a better cut, it may indeed be time to sharpen the mower.
Sharpen the mower
The easiest way to sharpen a reel mower, is by backlapping. You can pick up a backlapping kit, or easily create your own. All you need is...
You'll be using your crank to turn the reel, by attaching it to the place where the gear was removed from. You can either use a manual crank, or a drill motor and properly sized socket to turn the reel.
You can test the sharpness, by cutting a bit of scrap newspaper (be careful not to cut your fingers).
Once the mowers sharp and back together, get back out there and mow some grass.