What sort of permission, if any, will I need to obtain to use their comments in print legally?

You've actually just opened a really interesting can of Copyright worms! Generally, short posts (e.g. posts of "tweet" length) by users are not copyright (a common exception being a haiku); however, websites like Reddit generally have user policies about using comments obtained from their forums (like: you can't, for any commercial purpose). It really comes down to specifically what you are trying to do with the comments. You need a legal consult to discuss in full detail just what you are trying to publish! Depending on what you're trying to do, you may need Reddit's permission.

If there is no expectation of privacy, i.e., they are public posts you can reprint them without permission, IF, the material is deemed "fair use." Through the Fair Use Doctrine courts will consider principally four factors. 1. the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes; 2. the nature of the copyrighted work; 3. the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and 4. the effect of the use upon the potential market for or value of the copyrighted work. So, you must consider these factors in determining whether to reprint the work of others. Also consider that if it is a parody you are in a better position to reprint as well. However, every case is dependent on their particular facts. Consequently, I advise that you obtain counsel from a qualified copyright attorney.

You will need written permission from each person to publish their comments. Such permissions usually describe exactly how, when, and where the statement will be used and whether the author receives credit (full name, first name and last initial, etc.) and payment.

You will need to get specific permission from each person to use their quotes in your publication.

Do you mean quotations you are taking off the Reddit pages? That answer is probably in the Reddit Terms of Use. The Terms of Service may leave the Copyright on comment words in the ownership of the user, and it may not. However, you may be able to lift quotes off Reddit as Fair Use. The issue is if you are using them for the purpose of Analysis and Commentary. You would have to be actually writing the analysis and commentary. I am not sure you'd need to get any quoted person's permission if the Comments you are quoting are considered public, but if you are taking things out of context and mischaracterizing it such that it defames a person, you will likely run into trouble.The key is to be ethical. Use things in context, such as making sure your readers understand if the usual banter on a comment board allows the kind of talk you are quoting. For example, there are many online boards and chats where users regularly call each other names or issue death threats to each other, but the users do not consider these things serious. Your goal would be to show that in context. You should definitely talk with a lawyer. If I were the lawyer helping you, I would read your draft and then work with you to the final product. I would also warn you that you always risk backlash or possible lawsuits, but you can do your part to minimize your risk.