I can mean both, depending on the context.
Danish doesn't have progressive tenses, so your second sentence is incorrect.
As a basic rule in Danish, the names of instruments (f.eks.: violin )after "spiller"; and the names of professions ((f.eks.: vagt) after bliver & er are written without any article; neither the article of the repective gender of the noun nor the indefinite articles such as a/an are used.
"Jeg vil begynde at spille violin." = I will begin to play the violin.
Jeg vil gerne blive forfatter" = I want to become an author.