I live in Rechavia (in Yerushalayim). It's one of the more expensive areas, but there is an abundance of English speakers, and around Succos/Pesach time, half of New York is here. We have a renovated 1 bedroom and we pay ~$1200 a month (4200 shekel). You can find two bedrooms in the area starting at 4500 shekel, if you are lucky. Most will be at least 5000, with the really nice ones being 6000 or more.
Depending on how much you eat out, and if you are willing to buy Israeli brands instead of the expensive imports, you could spend four or five hundred dollars a month on food. Probably won't be too much less, but definitely could be more.
You can get on Meuchedet health insurance for ~$70 a month, and you don't even need a visa. If you get A3 visas, after about 6 months you are eligible for Bituach Leumi (national health insurance) which covers a lot more, such as childbirth. If you save your receipts for medical expenses before you got onto it, they will reimburse you.
We spend about $2000 a month, but we don't have a kid yet. I know diapers are expensive here though, so you'll have to account for that. Electric can be expensive when it's really hot or really cold out, but if you are careful it should only be around 450 shekel per cycle (6-8 weeks) max. Water is usually 200-250 shekel per the same cycle, but again it depends on your usage. You also have to account for arnona (yearly renters tax, the more your rent, the more you pay), vaad bayit (building upkeep, it's usually not so bad unless you are in a luxury building), and the usual surprise expenses you get anywhere.
Visas can be a difficult process, and I don't know anything about anything other than a student visa. If you are in a kollel, you should be able to get a student visa, and then add your wife and children onto that one (make sure you bring your marriage license, if you don't have one, then get one). There is a man in Bayit Vegan somewhere named Dagan who supposedly takes care of visas, Bituach Leumi, and some other things for you for a fee, but I have no experience with him.
There are plenty of Anglo neighborhoods, and as suggested above, you might want to take a pilot trip to find the one that suits you best.
Feel free to let me know any other ways I can help you out.