You've got three or four days to explore a new city, loads to see, and probably don't want to spend lots of money. How do you see everything a city has to offer without breaking the bank? A walking tour! Almost every city has one, but you've probably (and ironically) walked straight past it.
Every Spring I try and visit a new European city on a budget, but I always manage to see all the sights thanks to my love of walking tours. It may sound like something your parents would drag you on as a teenager, but if you're up for a walk and want to take everything in, it's the best (and cheapest) way to acclimatise yourself to a new city.
The best thing about walking tours is that they are free, you just have to tip your guide at the end. You also get to meet other tourists on the tour, which is particularly great if you're travelling alone. The guides are often students from the area studying history, art, architecture etc., so they really know their stuff! They generally say at the end of the tour that they accept as much of a tip as you think their tour was worth. I've never had a bad or even average tour guide, so I tip with that in mind: they are always so friendly and really do show you the best bits of their city. It will be the best value-for-money experience of your whole trip, just don't forget to bring cash with you for a tour tip; they don't carry around card machines! (Well, during my most recent trip, the tour guide actually did have a card machine, but it's rare.)
Though you can do your research online and book in advance, often the easiest way to do a tour is to navigate your way to the central part of the city, and just scout someone out. They are always open to chatting and finding out about where you'd like to go. Look out for signs in busy tourist areas, or pop into the tourist information office. Once you've met up with your tour guide, they will take you around not only the best sights, but also, hopefully, some of the lesser known ones. Most tours last around three to four hours. By the end of your first day you'll have seen everything "touristy" and now have time to go back to spots you loved and make time for other activities during your trip.
Some cities are obviously too big to walk around in one day, but there are alternatives such as bus tours, Segway tours (!), and bicycle tours. You can find popular walking tour guides online by simple typing the city you're visiting along with "free walking tours"; Tripadvisor also has lots of great suggestions.
So next time you're planning a trip, consider a walking tour as part of your sightseeing.