I recently moved from an old Nokia flip phone to my first iPhone.
You may find that your current SIM card will not fit in the iPhone but that's not a problem, it's just a matter of getting a blank SIM card and transferring your number to it. In my case I did this myself but your service provider should be able to do this for you free of charge.
I'm with o2 and after transferring to the iPhone, there was no difference in the way I was charged for calls and texts. In fact it was a lot easier to keep track of using the o2 app. Regarding data, Initially I was charged at £1 per day if I used it. Providing you only use data over wifi there is no charge.
I opted for what o2 call a bolt-on that costs £6 per month and entitles me to 500MB per month of data and 20 picture messages. So far this has been ample for my needs. What I like about the bolt-on is that it is in no way a contract. The £6 is taken from my top-up balance. They do not have my account details and cannot charge me any more than I have topped up.
Personally I would not dismiss apps in their entirety. There are some extremely useful apps out there e.g Google maps if you're lost