Certainly not taking away from the pros ... but I would start by getting manuals for them all (steve-pages or something like that has a TON of spoon manuals), and disassembling as per instructions in said manual. Give everything metal a good cleaning with a few shop rags and a degreaser of some sort to remove all the old gunk/grime/grease/etc. Clean it like its going to a concours. Then take a look at the metal and determine if it really needs refinishing or touching up.
Do the same process for the stock - wipe down with a tack cloth, then wipe with a rag barely dampened with some sort of (somewhat gentle) cleaner. Again, then decide on if it really needs refinishing, etc.
There are lots of "here's my process of refinishing this rifle" posts on rimfirecentral.com (all over, they post spoon specific not task specific although there are a few of those too), and in the gunsmithing/do it yerself section of
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Lever gun? Not my cup of tea, thanks... But if I had to go buy one in the next week or so, I'd probably look at the older ones (Marlin 39a, Winchester, etc), new production wise Henry, Browning. Maybe since you have a 10/22 a 96/22 ?
If you need an excuse to make a "rounder" collection, you are still shy a pump and a single shot break open.