so I have NO CLUE what I'm doing.

I just got an XBOX 360S for christmas, and I have a bunch of games. I already broke 1 game disk, and I want to back up my library so I can protect my investments.
So my question is-

Is there any new guide / relevant walkthrough on what my options are with ModChips or ODD (I'm not quite sure what an ODD is...) and how to backup and burn my legally owned games? Or can you guys just advise me on what to do?

I've modded and softmodded my Wii like 3 years ago. But I can't solder. At all.

I <3 Microsoft, and I don't want to pirate. That's bad. I just don't want to but the same game multiple times because my ******* sister decided to play frizzbee with my Kinect sports. Not the game, the disk. LOL

You might or might not be able to backup your collection. It depends on which DVD-ROM your console has. After you find that out, you can look into Optical Disc Drive Emulators like the xk3y, BoxZii, Wasabi, etc. If you want to flash your DVD-ROM (presuming it's flashable at this point), you can also do that. Check out the JungleFlasher tutorial for information. (Google is your friend.)

What's the difference between using an ODDE vs flashing my Drive?

And how do I find out what drive I have?

Flashing your drive will require making physical backups of the discs. ODDEs require just ripping the discs and storing the images on an external drive.

If you check the manufacture date/lot, you'll probably be able to guess what DVD drive the console has. This thread at Team-Xecuter is somewhat accurate.