Use the hyphens when the phrase is used to decribe a noun, but not generally after the verb. She attends school part time. (no hyphen). She is a part-time student.
The other style "rule" (and I use the quotations, because this is a style, not a grammar issue, and there are competing prominent style guides) is that you don't hyphenate with an adverb ending with -ly,
I'm going to leave the parts of speech part to someone else - I'm terrible at that.
But the same comments I made about the hyphens still apply. The style guide I follow says to NOT hyphenate -ly words, and you don't do it (regardless of whether there is an -ly) when it does not come immediately before a noun.