So... how do I install mods? Is it as simple as extracting files to the modules folder, or is there more I need to know? Thanks in advance for your help.

Mostly yes.

The "normal" mods (e.g. Armed Rebalance) consist of one folder you copy to ...\Mount&Blade\Modules.
If you extract the mod's archive and find a few dozen files instead (e.g. Blue&Grey 0.95), you have to create one folder, put all the files inside and move the folder to Modules.
And then there is the third type of mods (e.g. TEatRC), which consists of an .exe installator.

Then you just choose the mod you want to play in the launcher.

Thanks. I know I read somewhere that if a mod includes changes to items, I should start a new game. Does that apply to anything else?

It shouldn't be even possible to continue the "old" (native) game while playing a new mod. Are you choosing modules correctly?

In M&B, modules work as almost stand-alone game versions. That means you can't interchange native-mod nor mod1-mod2 savegames. Go to "... [User]\Documents\Mount&Blade Savegames". Each mod you have installed will create its own folder when you start playing (saving) it and will only be able to load its own savegames.

Anyway, you can never lose anything by starting a clean game. :)

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Disclaimer: I have currently installed the TaleWorlds version of the game (not the GOG) but I haven't heard about any differences between them in this. If I'm wrong, please, correct me somebody (who uses the GOG version).

It sounds like I misspoke. If I add anything new to an existing module (for instance, just yesterday I was working on restoring some character-creation options) is it a good idea to start a new file? (I mean, naturally I should in order to use the extra options, but hopefully you get my drift.)

Sorry, but I don't get what exactly does "restoring some character-creation options" mean. :/