If I restore & set up as new phone, can I save contact data?

Question

I've been searching for a while but still can't find the information I'm looking for...
I have an iPhone3G on an O2-UK contract. When I first upgraded to iOS4, the MMS camera icon disappeared from the left of the message box, and I was unable to send or receive any picture messages.
Then, 2 weeks later, when I reset the network settings the Mobile (Cellular) Data Network option (under Settings > General> Network) completely disappeared leaving me utterly baffled. I wasn't too concerned about it before but now I need to send some photos and it's extremely annoying.
I've been to the local Apple Store and the O2 Store twice and neither of them were ultimately able to do anything about it. They changed the Mobile Data Network settings multiple times and instructed me to restore the phone also.
On my latest restore I completely erased all data from the phone and restored from a backup, but the problem was still there... I know it can't be a hardware problem because BETWEEN erasing the phone and restoring it from the backup the MMS icon and network options all reappeared.
My question is this: If, instead of restoring from the backup, I was to bite the bullet and set it up as a new phone, would I be able to keep all my contacts and other data? Is there a way to transfer my contacts temporarily to the SIM while I carry out the operation?
It's extremely annoying that such a simple operation is so difficult to find... Why are contacts, notes and calender events not available to view and edit in iTunes? I can't see any reason why Apple shouldn't include this obvious feature.

Answer 1

Thanks for the advice wjosten. I haven't actually done the set-up-as-new-phone thing yet so it's not too late for me to do what you've suggested. There are still some problems though...
1. It says everywhere that contacts and calendar events are supposed to be synched with Outlook, however apparently Windows 7 Starter (for netbooks) does not come with Outlook at all. Can I get it for free somewhere or download some other program?
2. I've tried synching the ccalendar events with gmail, but it doens't work at all.
3. The only luck I've had in this regard is that I've managed to save about half of my notes to gmail, but another stupid little annoying thing which is bothering me is how to save all those un-backed-up notes in the "On My iPhone" account into the backed-up gmail account without moving each individual note one-by-one.