Daughter wants to visit Europe before she goes to college (Fall 2016), so I am planning a trip to Europe with my wife and 18 year old yr old in June - August 2016 time frame. She would like to visit Rome, Paris, Berlin, and London (may be Venice, Sicily too, if possible) in around 15-20 days.

. Dont know if it is possible to cover these places since we will only have 2-3 days at each place then. I would love to visit Greece Mykonos island if it doesn't add too much cost, but that can be for some other time, hopefully. This might be probably rushing it, but these cities are somewhat of a bucket list for the daughter and this is our only chance before she gets busy with college. This will be our first time in Europe and likely only time at least in these cities, so it is fine to see just a little bit here and there (at least cover the main things, time permitting) in each city.
Looking for your thoughts, suggestions, and of course, frugal ways to travel and deals you're aware of.

Flying out of the Sacramento, CA area, SFO/SJC are the closest big airports.
I currently have about 204,000 United miles, wife & daughter each have 19,500 United miles, 280,000 Southwest miles, 136,000 UR (wife's name), 75,900 AA, 157900 AMEX MRs, and about 278,000 IHG points. I will also be getting one free IHG night in Feb. Wife also has an offer for 50,000 United miles with new card. Till 2 months ago I was doing Serve and GoBank to earn miles (about 14,000 points per month for both of us).
Don't really care about flying business/first class, but it will be nice to have at least one-way trip in either of these 2 classes (of course would like to make best use of the miles).
I guess first priority should be to make flight reservations. I checked earlier today, and from SMF - LHR, I can get for about 30K miles on both United & AA (for June/July, but the problem is the return flights, almost nothing available till late August (I was only looking from London/Paris).
Looking forward to all ideas and suggestions for this long-time waiting trip. Would be great to stay in nice hotels. Have been staying in Holiday Inn (if available) whenever we travel, and we are fine with that. But, may be couple of nights in a really nice hotel, just to surprise the family

Anyhow, here I am seeking advice and suggestions, so I can start work on making it one of our best vacations.
Anything that I should be aware of, and how much should we budget per day/trip.