If I am out of my service area can I still read a book in iBooks on my iPhone? If not, is it possible to download part of the book and read it while I am out of the area?
Yes, iBooks can function normally while not connected to the internet. As long as you have the book(s) present in the iBooks app before you get out of coverage you will be fine.
If you are out of your coverage area and connect to another network, or roaming, you might have additional expenses for using another network.