How many watts would be needed to cover a 10 mile radius with good sound for music on the 88-108 MHz FM broadcast band? I am an amateur and for fun only.
This depends in your intended communication. Are you trying to hit a fixed receiver, in which case you can aim an antenna to that receiver? or are you trying to communicate with a handheld or other portable system? Are you on a portable system? Is the target system on a hill? The gain on the receive antenna, as well as the radiation pattern of your antenna makes a huge difference.
Additionally, the actual band used may make a difference. Many different bands permit FM operation, and some have unique properties that increase propagation distance.
Often, a 5w handheld with a rubber duck can hit a repeater ~10-20 miles away, but this is one particular example of possible communication.
As you mentioned that you want good quality for music, you'll really need a more powerful transmitter. Unfortunately, FCC part 97 (Amateur) rules disallow music through the amateur radio bands, so you'll likely need a commercial FM broadcast license to do that sort of thing.
I'm not qualified to talk at length about the particulars of antenna design, but there are other resources online. For broadcast FM radio, you'll need an antenna (a) in a high location, so receivers can clearly receive your signal; (b) high power, usually on the order of kilowatts. (WCBN-FM, a student radio station in Ann Arbor MI, uses a 200W transmitter, but has an effective range of around 10 miles).
This is a link budget question. Transmit power is just one factor: others are terrain, antennas, frequency, quality, noise, and so on.
If we restrict the variables to typical values for FM commercial broadcast stations, we can use the FCC's rules to make some estimations. By that reckoning, you need a transmitter of around 50kW with an antenna 150 meters high to achieve your goal of high-quality coverage over a 20 mile radius.
Unfortunately, that's a very expensive transmitter. And you need a commercial license.