You're probably seeing the SMS messages used for facetime/iMessage activation. You should not be charged for these if you are using a supported carrier and your account is properly provisioned for an iPhone.
As for data traffic, it will poll for e-mail and other notifications when in standby. This is normal. That's the point of a connected phone.
Thanks for your reply. Actually I didn't activate facetime or i Message, as far as I know. I'm pretty sure about that.
Regarding the traffic it happens that every time I make a phone call there's a simultaneus connection (few bytes) and then also during the night, when the phone is completed switched off, I've noted unsolicited traffic, also of several megabytes. This happens also during the day.
Actually I've avoided this problem by switching off "3G" or putting the phone in "fly mode" but in both cases I cannot receive phone calls or SMS because my carrier here works only in 3G mode, also for calls.
I'm thinking to format the Iphone again to the original settings without loading the backup, and then I want to see if this problem will stop. Have you or anyone any smarter idea?
If a few bytes here and there are a problem, then it sounds like a smartphone is not the right phone for you.
How small is the data plan that you have that this is creating an issue?
No supported carrier allows the use of the iPhone an any plan that does not inclued data, and the smallest plans around start at several hundred MB, while several GB is much more common.
Lots of things can use data. Pinpointing the exact source of a few bytes here and there is pointless and shouldn't be necessary.
Rather than turning off 3G, why don't you turn off cellular data if you don't want the phone using data?
Settings>General>Cellular. Turn Cellular Data 'Off'. Note that if you do this, you will not be able to send or receive MMS messages as they use the cellular data connection.
Pinpointing the source can be difficult. There are some apps in the app store that claim to be able to do so, but I can't vouch for how good they are.