TLV is the airport, which is not in fact in Tel Aviv. When it's not rush hour the drive from the airport to Tel Aviv is about 20 minutes, to Jerusalem about 40 minutes (the airport is between them on the main highway linking the two cities). The best way to travel between the two cities is by road (driving, private taxi, shared taxi or public bus). The trains are less frequent and the journey is much longer (it more or less following the road along the coastal plain, but then winds through the hills slowly up to Jerusalem so it can be scenic).
J'lem is by far the more interesting city historically, religiously, ethnically, and politcally. T-A is a modern city with great beaches and plenty of nightlife (though Jerusalem has its share) and historic Jaffa for a bit of history and diversity. From J'lem you can day trip to the Dead Sea by rental car, public bus or on a day tour. From T-A you can day trip by train to Haifa and/or Akko (aka Acre) which is one of my faves, and to Caesarea - though it's a bit more complicated with a short taxi ride from the train station to the sight. If you rent a car for 2-3 days while in T-A it would be easier to visit Caesarea (I'd still take the train to Akko/Haifa) and other sights on the coastal plain.
If you want to spend a couple of days seeing the area around the Sea of Galilee and the Golan Heights, you will need to arrange either a private tour or rent a car and DIY (not difficult but requires advance planning).
Driving in T-A and J'lem is miserable. I live in an outlying suburb of T-A and only go into the city a handful of times a year when I have no choice because of the traffic and lack of affordable parking. Now they've torn up some of the major streets in the city to build a light rail line, making it even worse. I drive a lot in J'lem but I know the city well and I know where I can park at reasonable rates (or even for free sometimes).
If you're staying in J'lem, getting to Bethlehem is easy. You can get a bus or taxi from outside the Damascus Gate of the Old City (easy to get to from the city center). It's about 3 kms so taxis aren't prohibitively expensive.
I think all the upscale hotels in T-A are within a couple blocks of the beach - doesn't much matter which part of the seashore - each has advantages and disadvantages.
May is a good time to visit. I think this year, the holidays are early so you won't have disruptions in early May when the weather is usually warm but not too hot. By the end of the month, it can get to sizzling summer temps but without the higher humidity that summer brings to the coastal plain.