"I need to get a new passport."
Translation:Jag behöver få ett nytt pass.
Some verbs need to put "att" between themselves and infinitive-form verbs; it is based only on the first verb. But there's no pattern, you just have to remember which verbs need "att" after them.
Yes, that would be fine. Jag behöver skaffa ett nytt pass (or skaffa mig). That will focus on the process of trying to get the passport, rather than on the result as få does. But get in English is ambiguous here, it has both those meanings.
måste is OK although it really means have to or must, not need to, but I know the difference in meaning between those has been lost in many versions of English (the same process is underway in Swedish, but it hasn't gone as far yet). But you cannot say får after måste or behöver – får is the present tense just like måste and behöver and you can only have one verb in the present tense. The second verb needs to be in the infinitive, so it would have to be måste få.