This is really going to vary by what cities you're visiting, but here's a breakdown from my trip to Vienna, Prague and Berlin last summer. It was me, my spouse and our two kids (4 & 6), but most of the things we did were free for kids; I've marked where they were an added cost.
Flights:
Actually paid with points, but equivalent tickets were $1,200 per person. I try to pay no more than $900 per person roundtrip to Europe by booking way in advance and going on off-peak times, but since my oldest is in school, that makes the off-season thing harder. (This is the first time I've gone to Europe in summer since a high school trip; all others have been January, April/May or October/November.)
Train tickets:
Vienna to Prague one way: $35 for two adults
Prague to Berlin one way: $85 for two adults
Berlin to Frankfurt one way, overnight: $275 (this was the cost to get a six-bed compartment reserved for the four of us; prices ranged from $50 per person if you were willing to sleep in a seat to much more if you wanted a real private compartment with a sink and stuff)
We looked into rail passes, but buying the tickets individually was a lot cheaper because we booked as early as possible. It sounds like you have a pretty fixed itinerary so you might be able to do the same. I think the real advantage of rail passes is when you're hopping from city to city at the last minute.
Accommodations:
Vienna hostel, 3 nights (family room with a double bed plus bunkbeds): $215
Prague rental apartment, 4 nights (2 bedroom): $390
Berlin 2-star hotel, 3 nights: $365
Food/Activities/Intracity travel:
This is hard to split up, because it was mostly cash, but it was $1,400 total for 11 days. Back of the napkin, I'd say that'd be equivalent to 3 adults, so about $42 per adult per day. We didn't do any really fancy activities like guided excursions, but some of the museum entries were nearly €20 each. We took public transit or walked everywhere; no cabs. Food was mostly from grocery stores (we usually had a kitchen available for breakfasts or to pack food), or little cafes or bistros that were recommended. We ate two meals that were over €40 total, and one that was €80 (but sooooooo good, we still talk about it); the rest were more like €15 at a street stand or cafe.
Other:
We had to get three new passports, which was nearly $400. (I was so mad...my passport was good for 5 months 2 weeks from the end of the trip, but Schengen requires 6 months. Also kids passports are only good for 5 years.)
I've never bought travel insurance; I think it's mostly a scam. You can often get refunds from even non-refundable things if you have a sob story.