“Invisible Children”, a documentary filmed by three American college students Jason Russell, Bobby Bailey, and Laren Pole, set out to Africa in search “for a story”,however, they uncovered the immense suffering of a civilian population in northern Uganda.

Should I put a comma after STUDENTS?

Also, "however" should I make that a new sentence or leave it as is?

At "however," you need to either figure out how to put a semicolon before the word or simply make it into a separate sentence at that point.

Also, commas and other punctuation usually go before the closing quotation marks. Here's my suggestion:

"Invisible Children," is a documentary filmed by three American college students, Jason Russell, Bobby Bailey, and Laren Pole, who set out for Africa in search of a story; however, they discovered the immense suffering of a civilian population in northern Uganda.

Read carefully -- I made several changes in there.