I've been to Europe several times - and budget is the ONLY way I know how to travel!
We've done a bunch of different things:
Booked package through GoToday.com . This worked out great in Paris, where we stayed at an awesome hotel in the first district (Mayfair Hotel) and the price was great and included transfers. We didn't do as well with this strategy in London, where we ended up in a pretty run-down hotel.
Booked hotel through Priceline. We did this in London and got an Intercontinental hotel for $100 US per night. Great hotel; great location; great deal. I stalked
www.biddingfortravel.com and
www.betterbidding.com to figure out how much to bid and which hotels I would likely get.
Rented an apartment. We did this in Rome through
www.vrbo.com . We ended up in a great residential neighborhood. Didn't end up using the cooking facilities (the kitchen was tiny and the restaurants are too good to pass up!) but got a lot of space for our money, and we had access to less touristy restaurants and cafes as we were in a residential section.
Picnics. We have had a great time buying local breads, cheeses, cold cuts, and wines to make great picnics in some of the world's most beautiful locations (like Versailles, Hyde Park, etc.) After all that restaurant food, we're usually happy for a lighter meal, and by eating a picnic lunch we can free up some of our budget for a bigger splurge at dinner. And it's also fun to visit the local grocery stores and experiment with the local language! We also make a lunch on our travel days between cities, since we don't care for the options in most train stations/airports.
Cheap European Airfare. There are small airlines that work routes within Europe that have amazingly low fares. Ryan Air b/w Ireland and the continent and Midland Express (I think that was the name) are a couple that we've taken. You can get one-way tickets very reasonably.
I also recommend visiting the message boards on fodors . com for some ideas of great restaurants. Many folks there are not on a budget, but you can also glean some good tidbits from others who are a bit more careful with their funds.
Have a great time!