I am first author on a paper about my research from last summer. Due to a busy school year and a bout of severe depression, I have taken FAR longer to finish this paper than originally planned.
When I send the draft to my coauthors, should I apologize? Thank them for their patience? Promise to be better in the future?
As a very young researcher, my primary concern is seeming professional and reliable to my coauthors.
In general, you should.
Especially if you are decidedly the main author of the paper, the delay itself should not be a major issue for the others (unless you have caused a "hard" problem, such as missing a deadline), but you may still have caused some inconvenience. For instance, your co-authors may have allocated some time for the paper that they couldn't work on due to your delay, and they might not have much time left now and have to squeeze it into their schedules again.
Whether the apology as such will be expected depends on the culture you're working in. Different cultures have vastly different stances on what "on time" means and whether it is expected in the first place.