I believe in cleaning lines, especially if you fish water that has lots of algae or other stuff in it.
I've used the SA fly line cleaner pad and the SA fly line dressing on my SA floating lines. Those work great. Just follow the directions on the products.
I recently acquired some Rio lines. Rio offers a cleaning cloth called "Wonder Cloth" and Rio has their own fly line dressing, "Agent X," I think it's called. These products work well, too.
One thing I learned on the Rio website was not to use things like Amor All on pvc coated fly lines -- they can speed up the "de-plasticizing" process. (Not sure I got the right word there -- they can speed up the aging process of the lines.) (Thirty years ago I used to wash my lines with liquid dish detergent and then treat them with Armor All to make them shoot better -- the dish detergent was OK, but the Amor all was wrong.)
If you go to the Rio Website, and go to care and cleaning of lines, they have some instructional videos for how to clean your fly lines.
And remember, the fly line dressings I mentioned are for use on floating lines. Just clean the sinking lines, don't put dressing on them.
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Storing fly lines: if you have a modern large arbor reel, you can just leave them on the reel.
Today's lines are supposed to resist "memory" or staying coiled when you strip them out next Spring. In the old days, I used to strip my lines off my reels and store them by wrapping them around a round Quaker Oats carton, it helped them resist the coiling effect. We can be happy that we don't need to do things like that anymore.
Storing Rods: what everybody else said. Main thing is, make sure they are totally dry. The rod tube or other rod case is the safest place.
Reels: depends on what you have. On my old spring and pawl reels, I clean them, dry them and lube them up before I put them away, preferrably in their own case. Modern reels with sealed drags and spindles that don't require lubrication, just clean them and make sure they are dry before you put them in their case or other safe place.