I'd like to visit La Alpujarra in Spain. There are few towns where someone can sleep.
What is the best base town to sleep and visit all the valley?
From your question and your comment I understand you want to stay in a town in La Alpujarra and use this town as "headquarters" to visit this area. Unfortunatelly there is no such town, due the Alpujarra is too big to go walking from any town to others.
You can use the GR7 (red line) to go from most towns to others in La Alpujarra. Have you considered the option of walk the GR7 from Lanjaron to Nevada in two or three journeys?
If you need to stay in a town, I'd recommend Trevelez, first because is the most attractive town in La Alpujarra and second because it has many interesting paths nearby. Also, it could be interesting the Alpujarra Trekking site.
If you want to walk between towns.. the closest to each other are Capileira, Bubión and Portugos.
If you fancy to walk more, there's a route called Sulayr which covers a very long distance, but there's a nice part of it between Capileira and Trevelez. More info about that route (and others) can be found here and here