I've had European vacations that cost a lot less than Disney ones.
First, travel at Thanksgiving, spring break or a time other then the summer. I've paid about half as much for spring break plane tickets as I have for summer.
Rent an apartment.
For a first trip, I'd suggest an entire week in Paris or in Barcelona. Go to the grocery store and buy breakfast food - yogurt, cereal and milk, rounded out by local pastries.
With kids, it's also terrific to have the option of eating dinner or lunch "in." Bring in sandwiches or even meals you can bake or microwave. Kids can get very tired and it can be a relief to eat dinner in the flat a few times. There's nothing worse than having your kids sleep through a "big splurge" meal.
Kids get a lot of price breaks at museums in Europe - sometimes free, sometimes at a reduced rate. I've learned to say my kids' ages in the native language (or you could write it down.) Sometimes the youngest would be free, sometimes the younger two, sometimes all three, sometimes we'd get a "family rate."
The public parks and other spaces are terrific in Paris and Barcelona...lots to do without paying entrance fees.
You can also save money with subway passes (rather than individual tickets.) Research the public transportation options once you settle on your destination.
I've taken my three kids to:
Paris
Italy (Rome, Florence, Venice)
Barcelona
Germany
It can be done!