Is it worth upgrading my companion's gear in Skyrim?
Solution 1:
The game will decide for you what they should wear: they'll always try to equip the best stuff they have. It is worth giving them upgrades, especially if you don't plan on getting them killed.
One thing to keep in mind: you can't teach an old follower new tricks. If your follower thinks, acts, and combats like a mage, giving them a two-handed axe or heavy armor isn't going to be at all useful for them because they aren't going to get better at axes or wearing heavy armor, even though they will choose to use it in lieu of magic.
Solution 2:
There seems to be a limit to how much you can upgrade a follower's bow, as I found out the hard way.
I made a Legendary Daedric Bow and gave to my follower (which was an upgrade from her Legendary Glass Bow) and she instead skipped it and started using the Hunter's Bow she started with. I believe the highest bow combo an NPC will use is Glass Legendary + Iron Arrows — any higher combination and they will default to the lesser Hunter's Bow. I hope to find out more as I experiment.
Solution 3:
Even worse, some followers--especially archers--don't have certain skills, like heavy armor, block, or two handed. But, these characters will equip and use these items anyway if you give them to them.
I've solved this by smithing and enchanting nice items of the appropriate type. This seems to solve the problem for a quite a while, until I accidentally asked him to carry an iron dagger for vendor fodder--even though he had a legendary enchanted bow--and watched him charge a giant. facepalm
A small note--it doesn't seem that giving soul gems and a soul trapping weapon to a follower works for collecting those low level petty souls that these characters constantly kill before you can get a shot off. The effect goes off, the animal dies, no soul gem filled.
Humorous side note: the ragdoll effects of watching your tactless follower fly hundreds of feet into the air after being crushed by a giant are amazing. Serves him right.