My apps I use are :
Pure Calendar : it's a nominal fee, but it really gives you quite a few options on how to display your calendar widget on your homescreen to keep you up to date.
Toggle weather widget : I just like the way it displays the time and the weather on my main screen.
Facebook : It's Facebook
Twitter : It's Twitter
Lookout Mobile : Runs in the background and protects you when you download files, plus has some great security features built in should your phone get lost or stolen.
Advanced Task Killer : Helps you manager your tasks
Astro File Manager :
Bump : Great app, works with the iphone's bump to exchange contact info quickly with someone else.
Root File Manager : Gives you complete access to all files on your phone if it is rooted. (If rooted, you'll also want Superuser and Boot Control as wella s CacheMate)
Pandora - Really stable way to listen to music (watch out if you don't have unlimited data plan, or this could really add up).
Ring Droid - Lets you edit your music to create ringtones for free (Mabilo ringtones has quite a few ringtone options, but it is no longer on the market so you'll have to download the .apk file from www.mabilo.com
unless you have AT&T).
All of the Google standard apps - Voice, Sky, Earth, Maps, etc
Google Navigation - Decent FREE GPS service that you may have to download (if you're on AT&T) from the market
Handcent SMS - A little easier SMS/MMS texting client than the standard Android one
Mileage - Tracks my car mileage
Runkeeper - Keeps track of my running
All of these have been stable (on my G1 and Captivate) and i use them all. Hope that helps some; good luck!