This approach is usually good for a major city. You’d book the air and hotel from one of the same travel companies that you used for guided tours. Some offer choice of hotels at different locations for different prices, so you’d have to read up a bit and try to pick a hotel at a convenient location or just go with the lowest cost one. After you book, remember to get the visa, if you need one.
You’ll be exploring the city on your own, so you also must take care of the transfers between the airport and the hotel. You should search and see if there is a train, metro, or bus available for the trip from airport to the hotel and take it, if it’s available.
To plan for sightseeing on your own, you should get a guidebook for the city and study it a bit before you depart. If you are unsure about what to see, a good option is to take a half- or full-day city tour and get an overview of the major attractions.
Mastering the local public transportation options is also helpful. We usually take metro or tram as much as possible.
You should develop a rough itinerary as well. I typically divide the day into two halves and just allocate activities to each half-day, along with times for lunch and dinner. Remember not to cram in too much in a day. If possible buy tickets online (before departing on the trip) for any museums or other sights that require tickets. That way, you can avoid any long lines for the tickets and save some time.
That’s about it. With a little bit of peparation, you should have a great time exploring the city of your choice on your own.