I need to do some work under my bathroom sink. While there, I am thinking of replacing the old hot and cold water shutoff valves. (See photo.) What is there does not seem very high quality. Are there shutoff valves I could use that are higher quality, perhaps ball valves?
You can certainly get 1/4-turn (ball-type) shutoff valves.
They are arguably better than screw-type valves for the service (they generally do well with being left for years and then used.)
Then again, it's impossible to judge the "quality" of your valves from a picture - if they operate correctly and fairly easily, there's no particular need to replace them. The wall trim is corroded, but not part of the valve and presumably out of sight under the sink cabinet most of the time.
If it ain't broke, don't fix it.