"I need to get a new passport."
Translation:Jag behöver få ett nytt pass.
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.
"Få" is the verb meaning "get". Whether you need the "att" before it depends on the verb before it. Here's a good thread about it: https://forum.duolingo.com/comment/7075383
This the same as "to" before English infinitive verbs. You would say "I want TO go to the store", not "I want go to the store." BUT you would say "I must go to the store", not "I must TO go to the store.
So just like in English, you just need to learn how all these verbs work.