From what I've heard, you basically add some effect to an object, and then it runs out with use, based on soul stones. So I guess it works kind of like poison - you apply it an then it runs out. Do I have the right idea? How common are soul stones, e.g. How often can I recharge my enchantments?

Kind of only it can Be recharged. Rarely common but not too common.

You can recharge them as often as you have soul gems.

kinda common in some dungeons with mages, apparently there is also a thing where you destroy enchanted things to get the enchantment to add to another thing.

Basically it works like in oblivion, look up its enchantment page for a lot of info.

The changes are that now the effects are separated from magic spells you know. So even if you know all flame, frost and shock spells in the entire game and can use them all you won't be able to use it when enchanting an item. Breaking down an enchanted item does add the effect from that to your enchantment list.

Only weapons have charges, clothing and armor will never need to be recharged. The stronger the effect you have enchanted to the weapon, the more often it needs recharging. The stronger soul gem you used when creating it determines how powerful enchantment you can do and how many uses it has. A petty (super small) soul gem could allow you to hit 5 fire damage 20 times before being empty, recharging takes 200 soul power (equal to 1 petty soul gem). A grand soul gem (super big) allows you to hit 10 fire damage 60 times before needing recharging, recharging takes 1600 soul power (equal to 1 grand soul or 8 petty souls).

*note, that 200 and 1600 soul power were only figurative numbers that are most likely not what they are in the game*

i am wondering if this only happens to weapons - e.g. "draining" out with each hit, or also to armor? e.g. an armors effect being for lets say +20% sneaking being drained after 1h of sneaking or so...

I do not have answers (though I can describe how it worked in Oblivion, Morrowind, or Daggerfall) but I do have a question to add:

Is it like Oblivion where recharging is only needed for weapons/staves? Or is it like Daggerfall were even while-worn enchantments run down over time?

Anyway, to the OP: In Oblivion soul gems were not too hard to get, and there was a single re-usable soul gem available. And weapon enchantments typically lasted for more hits than poisons (actually poisons only lasted for 1 hit in Oblivion, but that's changed in Skyrim obviously). There was no limit to how often you could recharge a weapon, just to maximum charge it could reach. Each enchanted weapon would have a maximum charge (depending upon the soul gem it was enchanted with) and a charges-per-use (depending upon the enchantment effects). Recharging one would increase the charge by an amount dependent upon the soul gem used to recharge it, up to a maximum of the weapons maximum charge.

I hate having to recharge enchanted items. I think you should use the soul gems to enchant the items but you shouldn't have to worry about the damn charge going out in the middle of battle. I think it also forces you to have to use soul trap. I just don't like it

Let me explain how Staves work:

Damage done is static and depends on the strenght of your enchant and using them increases your enchanting skill.However your skill in school of magic decided how fast will those charges be used. If you use Staff of Firebolt it will use something like 100 charges if your Destruction is low, but if you maxed your destruction it will use something like 25 Charges.

@dampas thanks, thats good info

So the charges on the staff go up based on your skill in destruction, damage remains static.

Im guessing this is based on destruction skill level alone and does not require perks in destruction?