Fly Round Trip.
I started scoping out flights months in advance, and what I noticed is that round trip flights to Europe were way cheaper than flying to one country and then home from another. So I booked a round trip to London, which only cost about $600 per person (including taxes and fees). The trip started in Paris, which meant getting on another (separately booked) flight from London to Paris once we arrived, but then I ended our trip in London for easy access going home.
If you are investing in a $600 flight to Europe, my strategy is to "see it now, while you are here" to get your money's worth. The only catch? The longer you are away from home, the more money you spend on food and other expenses, so it's all about finding a balance between the two. Overall, I wouldn’t opt to fly to Paris for just two nights if you can help it, though – might as well stay longer if you are spending the money to fly across the Atlantic!
Visit Fewer Cities.
Every city I added on to the trip added another major expense. One-way flights and train rides between cities in Europe are often only about $100 per person, but they add up quickly. Looking back, I wish I would have skipped Milan and just gone to three cities on this trip to save even more money.
Use Public Transit.
Another huge money saver: We took public transportation, not only for sightseeing, but also to and from the airport. This means lugging your own suitcase up and down the stairs on the subway, fighting your way into a hot, crowded subway car with a big suitcase, etc. – but you will save hundreds of dollars this way. (I kind of enjoy the challenge of navigating public transit, so this part was fun for me!)
Stick to Free Stuff.
Pretty much every historical attraction has optional fees. You can pay to go inside Westminster Abbey, the Colosseum and Notre Dame, and while all of those places are amazing, at the end of the day you will be spending hundreds of dollars if you go into every single one of them.
We chose to explore a few, but for the most part there are plenty of free things to do, so paying is really unnecessary. In London, I recommend skipping the huge fees at Westminster Abbey and focusing on the totally free (!) museums throughout the city. In Italy, all churches are free to enter including my favorite, St. Peter's Basilica. In Paris, the Louvre was worth the extra money, but I opted to tour only the free part of Notre Dame.
Get a VAT Refund.
If you do some shopping in the U.K., save your receipts! You can get a refund on taxes at the airport by showing them the item and the receipt. Look for signs when you get to the airport for the VAT refund office.