Hi Ken. How good is the shape you're in? You'll be walking a lot, some of it up and down slopes and steps. There's plenty of time to get in better shape before February. February in one of my favorite months here because of the mild weather and relative lack of tourists. The mild temps make sightseeing much easier but the rain can make paved walks very slippery. Be sure you have comfortable, thick-soled shoes with good traction. There are usually lots of spring wild flowers by mid-month.
It's probably not too early to make hotel reservations. While prices are lower in winter, overall prices are higher here than in Europe - which no one here understands.
Try to find a 2-3 night guided tour based in the Galilee. That's where many of the Christian sites are located and day trips from the big cities will spend hours of bus time just going back and forth. Driving in the area isn't difficult, the roads are good, the signage is also in English and there's not much traffic. The only drawbacks I can think of are that a couple sites on the Sea of Galilee are a tad difficult to access and have very limited parking. Do not drive to Nazareth. If you want to go there, go on a tour! Driving is a madhouse and parking is almost impossible to find. You can hire a guide/driver or probably find a group day tour. Haifa would be a better place to base for that. It's also an easy trip from Haifa to Akko (Acre) which has wonderfully preserved Crusader sights.
I can't think of any Christian sites in or around Tel Aviv. Do plan to spend a lot of time in Jerusalem, which is chock-full of them. You can easily see most on your own. You absolutely do not want a car in Jerusalem. The one-way streets and dedicated bus lines would drive you crazy even without the many traffic jams - and parking is expensive.
You cannot drive a rental car to Bethlehem, it's illegal and there is a checkpoint which will not let you through. Cars, taxis and buses registered as Palestinian can go there. It's easy to take a bus from East Jerusalem or a taxi or get a private guide/driver.