Cyanogen OS system upgrade

I've rooted my phone using a guide on this forum. I have also installed TWRP recovery.

OK . you can then download the official firmware and install it. When prompted by TWRP recovery to install SU chose NO.

Should I install everything in the zip? Won't that wipe all of my data?

The cm-12.1-YOG7DAS2FI-crackling-signed-81d046434c.zip updater-script does not seem to wipe data.
You may want to wipe dalvik and /cache first though.

OK, one last thing: How do I flash the zip? Can I just put it on my phone and use TWRP to install it? Or do I have to use adb? And what's dalvik?

You flash using TWRP. The zip can be put on sdcard (internal or external).
The dalvik cache holds some sort of compiled java code. For minor upgrades one can perhaps leave it untouched. For major upgrades one typically wipes it. It does not hurt to wipe dalvik cache. After install, it is regenerated as part of the android upgrade.
Note that stock firware is probably unrooted so you may want to reinstall supersu update zip from twrp. Also, upgrade may or may not overwrite twrp with stock recovery so you may have to reflash twrp.
I have not done the upgrade myself (I will one day, I have no Swift, a friend has one).
Success

Ok, I flashed the update, everything works fine except that every few seconds I get a box that says "Unfortunately, Google Play Services has stopped.". Super annoying!
EDIT: Allowing wakelocks for Google Play Services in Privacy Guard fixes it.

@seb5049
Just for my information: did you wipe dalvik-cache?
Google Play and Google Play Services typically may get updated change during or right after upgrade.