Your phone numbers/calendar should be backed up to your gmail account, as long as they are Google Contacts/Google Events. You can easily check this by logging into gmail on a pc, and seeing if the contacts are there.
Apps that you paid for can automatically be re-downloaded if need be. Free apps are obviously free and can be re-downloaded. If you want to backup an app's data, like back up saved games for Angry Birds or back up mileage info in aCar, you would need to be rooted, and use Titanium Backup. That's the only way. You can use AppBrain to back up a LIST of apps, but that wont touch an app's data.
As far as settings, Titanium Backup can do that too, with root access of course. However this cannot be done across roms. You can also make a Nandroid backup with root access, which is basically just a clone of your phone's current setup: apps, settings, data, everything.