After some extensive searching, the only solution I have found to this problem is that you have to save styles with templates. It is not possible to save stylesets as you do in MS Word.
This means that you have to create a new document, set up the different text styles, and then save it as a template. The next time you create a new document in Pages, you will find your template under "My templates".
I have sent a mail to the developers, and hopefully they will change this in the future. Sometimes you want to change styles after you have written the document, and this is not possible in Pages without a lot of copy-pasting.