What would be the ideal way to implement this sort of thing? The idea I have in my head right now is to have a comments table and have each comment have a thread identifier
Look at Managing Hierarchical Data in MySQL, specifically the section called "Nested Set Model". You may have to read through it a few times before it makes sense (I did) but it's worth it. It's a very powerful way to work with nested data and retrieve the parts you want with only one query.
On the downside, for updates you have to do a lot more work.