For rods, wash the grips to remove dirt/scum, clean the blank and guides with a soapy paper towel. Check for broken/cracked ceramic guide/tiptop rings by wiping the ID with a Q-tip. If one is cracked it will snag the Q-tip. Store straight in a not hot room. Hanging from tip or a guide works fine.
Reels:
Wipe clean with a damp soapy paper towel, followed when dry with a slightly oily rag like you would for a firearm. Oil all external moving joints, like grips, bails, level wind pawl. Not a bad idea to brush the pawl and worm gear clean with mineral spirits or other cleaner, then oil it. (I like reel butter, but any good oil like 3 in 1 or sewing machine oil will do fine). For the inside I don't recommend taking the reel apart beyond what you see when you open it up. Reels are too complicated for me these days, and unless it has been dunked, you don't need to go further. If it's sort of gunky, brush to clean with mineral spirits, blow it out with compressed air if you have it, and relube. Pivots, sliding contacts, etc with oil, gears with reel grease. Look at your reel schematic to be sure you have found all the bearings. Don't overdo it, just a drop is usually sufficient. If the reel has been dunked, send it to a good repair person for a pro job. DVT is good and discounts forum members. I don't recommend servicing the drag unless it's not working smoothly. If you do service it use drag grease, not regular grease. Be sure to lay out the parts in order so you can get it back together properly.
Loosen the drag before storing. There are many good tutorials on line, both pics and UTubes. Do a search and you will find many.