It would be lär dig själv svenska, it's only either lära sig 'learn' (reflexive verb, the reflexive particle changes with person, so jag lär mig, du lär dig and so on), or läsa as in 'study'.
You do need själv to make it mean 'teach yourself', without själv it would most likely just mean learn*.
Jag har bara lärt mig svenska i en månad
Swedish doesn't have the progressive, so "learn" and "am learning" both translate to the same thing. This is true for other tenses as well - so rather than "have been learning", you'd say "have learned", i.e. har lärt [mig].