I have backpacked by myself across North America six times now since the early 1990s and loved it. Last year I went to Moscow and London for 3 weeks and that was fantastic, too.
I would usually plan one or two things to see or do during the day, and perhaps take in a discounted show or a movie in the evening. There are a lot of bargains online if you plan ahead.
There is a fun in making a fairly loose sight-seeing itinerary, but also going where the day takes you if you desire (without upsetting others if you're traveling with a group).
I am a better traveler alone. In group situations, I'm usually too worried if the other people are enjoying themselves (because it is such a big financial commitment) to enjoy what I'm seeing or doing instead.
Most of my friends are put off by the idea of doing it basic and staying at youth hostels, but I love it. All I need is a bed, a locker, and a shower, and I'm set.
I would rather spend money on attractions and gifts, than expensive hotels that I'll only be in when I'm asleep. I did this once on a trip to Sydney and I was never there!
Good thread, by the way.