Explore on foot - This is a great way to get the feel of a city without spending a dime. You can walk around and gaze at the buildings and watch the locals. My favorite city for walking is Venice. I love getting lost in the alleyways and coming to a dead end at the canal. Plus in Venice, you never have to worry about cars or speeding taxis!
Visit a church - Entrance is usually free and many European churches were designed by well-known architects or artists and have paintings, murals and statues done by famous artists too.
Eat at local markets - Many cities have local markets (some of which are quite large: Mercato Centrale in Florence, for example) where you can buy anything from fresh fruits and vegetables, meats and cheeses, just-baked bread, sweets and wine. Getting a few things at the market and having a picnic is an inexpensive way to eat any meal while traveling.
Aperitivo! In Italy, aperitivo starts in the early evening and ends around dinner time (which of course in Italy is late!). When you order a glass of wine at a bar, the bartender will also bring little plates and bowls filled with things like olives, crunchy potato chips, small bites of meats and cheeses, crostini and miniature pizzas. Sometimes (usually) this can be enough to last you all night. Also in Venice, we love to eat cicchetti which can substitute for a larger dinner. Most items (think marinated vegetables, assorted crostini, meat and cheese, calamari) are around 1 euro, so you can easily get 10 things to sample for dinner. Our favorite spot for cicchetti is Do Mori. Plus: how fun would this cicchetti tour be? I'm dying to try it.
Buy a combined ticket - If you know you want to see the major sites of the city, buy a combination ticket that can get you into multiple museums or tourist sites. Some of these tickets even last for several days so that you don't have to hit everything in one (long) day.
Rent an apartment - In April, when we went on our great tour of Italy, there was no way that we could afford to stay in hotel rooms for a month. So, we decided to try airbnb....and we loved it. We stayed in apartments in Cinque Terre, Rome, Bari and Sorrento for a fraction of what it would have cost to get a hotel room. Plus, all of the apartments we rented had kitchens so we were able to cook meals at home rather than spending money at restaurants every day.
Ask a local - If you can get a restaurant or sightseeing recommendation from a local, take it! Most spots that are less touristy are also less expensive.
Drink locally - This is especially easy in Italy where each region has different and amazing wines...and when we were in Prague, we ditched the wine for local beer. Find out what's made in the region and stick with that; it's always less expensive and better.
Travel off-season! This is often easier said than done since people love to travel in the summer or around holidays....but you will save tons of money on flights, hotels and even restaurants if you go during less touristy times. Example: November. This works well especially if you're heading to a larger city - everything will still be open but you won't have to deal with the crowds, heat and prices of summer.