What I couldn't live without might be stuff that someone else couldn't care less about. It's all about personal preference really.
I personally use:
Handcent for texting
Dolphin HD for browsing
K9 Mail for email
Facebook for Android in lieu of the FB for HTC app
SiMi clock for my clock/weather widget
Weatherbug
AndFTP for FTP'ing in to my home laptop
ES File Explorer/Astro File Manager for managing on-phone files
AudioManager for managing various volume levels
PowerAmp and MixZing for music playback
Pandora
Titanium Backup for, well, backing up (requires root!)
Root Explorer for modifying system files
RealCalc calculator
Engadget news widget
drocap2 for screen captures (requires root!)
System Monitor for graphically keeping an eye on battery usage
I guess that about does it for most of my personal must haves. I have a ton of other apps that get sporadic usage, such as IMDB, Zillow, games, and other stuff. As far as backing up all these apps, I use Titanium which does require root access, but I've heard of appbrain as well and I don't know if it requires root. I do know though that you can't save game data, just the app itself. And you also can't uninstall bloat without root, it's a necessity!
Again, it's all in what YOU want! I recommend just going in to the market and exploring, that's half the fun

Oh, and DO NOT install a task killer! It'll do nothing but speed up battery drain, as it goes against how this OS is set up to work.