I would recommend getting a car. The public transport links are alright, but a car is MUCH easier, only be aware - Israelis drive like maniacs. As it's a relatively tiny country (four minutes of air space), you can stay in one place and then do day trips, as most places are about a 2/3 hour drive max from Tel Aviv (apart from Eilat, but I wouldn't recommend it).
Go to Caesarea. There are Ancient Roman ruins and they have a children's activity centre (though I've only been there in the evening to go and eat).
Naturally, Jerusalem. I would recommend strolling around the Old City which is fascinating. You can only access the Al Aqsa Mosque early in the morning (if you're not Muslim), but the other major attractions (Wailing Wall, Holy Sepulchre etc.) are easily reached and entered into. Because they're religious sites, you should wear more modest clothing (so no bare shoulders or shorts). For the wailing wall, if you want to go up, men and women are kept separate (but you should be able to see your DS through the separator). One of the best tips I as given is to go to the Austrian Hostel (which is on via Dolorosa) and go up to their roof. DP and I just strolled up without anyone hassling us. It has the BEST views over Jerusalem - it's incredible.
There's also Akko (I haven't been, but DP went recently and said it was amazing) and near the Dead Sea (also pretty good) you have Massada which is incredible and has lots of Roman history attached. Be warned though - it gets incredibly hot during the day and you should take water with you (you can fill it up while you're there). You can either go up using a cable car (which is relatively quick) or walk. I have never walked. I'm too much of a wuss.
Tel Aviv itself is fabulous (I'm a big fan). Lots of nice beaches, relatively safe and clean, amazing food ... very friendly. If you go to the south you get to Jaffa (which is the old City) which is also fascinating to walk around. One of the best things in Tel Aviv is Tamara, which is a frozen yogurt shop on Ben Yehuda (which runs paralel to the beach). For 15 shekels, you get a frozen yogurt and as many toppings as you like (fruits, mini meringues, chocolate pieces, buts) and hot chocolate sauce. It's INCREDIBLE. I've seen similar places opening in London, but none compare.
A lot of Israelis speak quite good English (particularly in the cities), so that shouldn't be too much of a problem, though there seem to be more in Tel Aviv than in Jerusalem.
From a safety perspective - be ready to be quizzed a LOT in airports by security staff. I had a half hour of questioning once by El Al staff at Heathrow, but this is par for the course. Don't let it put you off. Things have calmed down a bit over the last few years, so it doesn't feel as tense and Israelis are so used to it, they're completely oblivious. the first time I went was during the Second Intifada, and a couple of days after I left there was a suicide bombing in Tel Aviv - scary. There is a lot of security (you have bags searched entering malls and events etc.).
Have a great time! The politics side is very complicated