Welcome!
My favorite app for music syncing is iSyncr.
For file transfer in general I LOVE Wi-Fi Explorer.
You can't backup the ENTIRE phone unless you have rooted your phone.
In the meantime:
Gmail will automatically backup your contacts, calendar, email, and anything else you've checked in the accounts and settings screen for your Gmail account.
Any apps you've purchased will always be available to you in the Market. The key is that you always sign in to the same account.
As for app settings, these are not natively backed up unless the individual app has a settings backup feature built in. Most settings intensive apps do have such a feature.
There are free apps in the Market that backup call logs, SMS, and MMS messages.
If you're rooted then you can use programs like MyBackup Pro to backup all the stuff I said above plus settings. Some people swear by these programs. I personally have spent more time debugging issues I've had after restoring a backup than if I'd just started with a clean slate.
In laymen terms rooting is a process that results in you having access to areas of the file system that are normally locked from user access. It just so happens that this area includes all apps (including those that you install to internal memory) and their corresponding app data/settings. Backup programs need this level of access before they can access this data.
Check out the General Help & How-To section as well. Milominderbinder, one of the AC Advisers, has spent the last year writing up Guides that would be a dream to anyone new to Android.
P.S. don't use task killers! A puppy dies every time you do.