Things that I have found that make the double action harder than it should be all have to do with the cylinder and "miscellaneous crap" buildup in the area around the ejector rod, on the yoke. That's a technical term, btw. Disassembly is easy, and you should not be intimidated. Reassembly can sometimes require 2.5 hands, but it's not all that hard once you get the hang of it. The most difficult part will be the smaller springs and avoiding escape velocity when reinstalling those. The mainspring is a cinch with a small pin used to capture it.
In my opinion, Smith has one easy to disassemble revolver there. I break my 637 down every time I have a long shooting session, as I shoot cast through it, but I don't yank all the guts out of the gun. It's mainly just a cylinder breakdown to get the lube, lead, and carbon out of the parts that are supposed to spin. As mentioned, the internals are best dealt with with some chlorine free cleaner of some sort and liberal finger pressure on the nozzle, also known as the "shoulder thing that goes up." Dental picks and the nylon versions are great for buildup in these areas.
A couple things to remember in all of this is that smith cylinders are reverse threaded and that excessive force is not your friend in the disassembly. I have had a few Smiths that have red loc-tite on the side plate screws and yoke screw. Other than that, there are no real problems. Make sure you use a well-fitting screwdriver blade for those screws. Of course, if you bugger them up, add some more to your next Midway or Brownells order if you're worried about the unsightly appearance. Buggered screws and side plates are a good indicator of bubba-smithing and will hurt the gun value next time.
Since you've gone through the trouble to take it apart, lube it up in there as well. You don't need it to be leaking oil into your pocket holster, but there are a few areas I like to add a little grease, etc. There are some good videos on youtube, as I recall, on where to apply some lube.
Also, I like to have a dremel on hand with some nylon or brass brushes for certain areas for cleaning. You don't need warp speed, and an adjustable dremel is the way to go here. I use one to get the carbon from the top strap and sometimes on the yoke/crane.