Yes, you can do Europe on a budget! I went for two weeks in 1995 with my mom, sister, and her two kids. We went back again for three weeks in 1998 and took my dad too - and paid for him & mom as their anniversary present.
Rick Steves has a book about Europe with kids (I think it's him - I was watching the show on TV some I'm sure he wrote a book about it! lol). Do lots of research. We liked to visit out of the way places, not just the hot spots. We packed dark color (don't show dirt as much) loose clothes that could be layered (denim jumper, overalls), and wore very good hiking shoes. You're not going to be seeing the same people every day, so no one knows if you alternate the same few clothing items for several weeks. We also had a little clothesline and a small package of soap so we could do laundry and hang to dry in room. Oh yes - we brought our own toilet paper.

You can photocopy pages of pertinent info from travel books - then toss the page afterwards to lighten your luggage.
Check the train passes for good deals, even for multiple countries. We rented a car in Germany once, and dropped it off in Paris. Learn the bus schedules of where you are - not all places have buses that run all the time, and many stores close at dinner time - some even close for lunch. We loved the little grocery stores and would pick up lunch items and have picnics. Check for local festivals, fairs and holidays. Watch where the locals eat - and go eat there.
For the buses/trains - if you have slower walkers, make sure you try to get on the next to last bus/train - that way, if your slow person gets really slow, you won't miss the last bus and have to take a taxi. I carried a little spiral notebook and wrote down departure & arrival times so we wouldn't forget.
We stayed in a time share for part of the time, and small hotels the other time. Felt lucky to get a room with a bath/shower on our floor, lol. There weren't many public restrooms, and many times they have attendants who want to be paid, or the doors have money slots on them.
We got tapes from the library on learning the languages so we could try to order food, and steer clear of stuff we didn't want to order; we used traveler's checks; save room in your luggage for souveniers; back packs are a necessity; also used those body belts to hold money & passports.
It's fun planning. Have a great time!