Leaving from a major airport in the States, spending a total of 2 weeks overseas, a little over a week in the UK and about 5 days in France and Italy combined. Staying in nice hotels not camping or hostels (not overly fancy, but clean and comfortable, equivalent to Marriott 3 star standard for instance)
How many people are going? The cost of the flight depends on the airports involved, and when you want to travel. On average, I would say $800 for a roundtrip flight (can vary from $600-1000). You can find exact prices using a flight comparison engine:
http://momondo.com
http://skyscanner.net
As far as hotels, it depends whether you’ll be in major cities like London/Paris/Rome (which can be a bit more expensive, and you have to be concerned about location) or in smaller cities.
Expect around £70 for a cheap but decent double room in a hotel or b&b in London. 2-star hotels (which can be nice if you do your research) in Paris and Rome usually cost at least €80 for a double room. Tripadvisor is great for finding hotel rooms, just search a city and set the sliders to your budget and above a rating of 4. The results will be the most inexpensive Hotels with the best rating. So that’s $120 (current exchange rate) x 14, or $1680 for two people. A single room can be 20-30% less.
http://tripadvisor.com
Food/attractions/shopping depend on what you want to see and do. You can get pub or cafe meals for under $15 per person, a restaurant can easily be double that or more. Alcohol costs are higher than in the US. $30-50 would be my minimum recommendation if you plan on eating out (cheaply) twice per day. Add $5 per drink if you’re going out, and $15 (varies) for attractions. $700-1100 per person would be reasonable.
So for two people, I’d recommend at least $5000-6000 for the whole trip, depending how frugal you plan on being. You’ll need more if you plan on visiting more than one city in each country. You may get by on less by finding good deals (or more if you plan on shopping or going out a lot), but it’s best to be prepared.