Learn Dutch: Je gelooft me niet. - You don't believe me.

Yes, you can! Mij is the stressed form of the word "me", while me is the unstressed form. You would use mij if you want to add heavy emphasis (like saying "You don't believe ME!"). In everyday, normal speech, the unstressed me is the most common choice.

Why does niet come after me instead of before it?

In Dutch, the word niet (not) usually gets placed after a specific object. Because me is a specific object pronoun, niet moves past it to the very end of the sentence.

The verb for "to believe" is geloven. Why does it change to an 'f' in gelooft?

Dutch words almost never end in a 'v'. When we find the basic stem of geloven, the 'v' changes to an 'f' to make the sound sharper (giving us geloof). Then, because the subject is je (you), we add the standard -t ending, resulting in gelooft.

Could I start the sentence with jij instead of je?