What is Reddit's opinion of Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science, Series Number 43)?

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I'm interested in why you want to learn these specific things. Is it for computers? In any case, the best thing that will help you learn proof theory and temporal logic is probably getting good at math since a lot of what you would encounter is pretty technical. That book you linked to looks like it would be good for that. If you want to learn about proof theory this and this look like a good start. But keep in mind they are both rather technical books. As far as temporal logic goes, it's probably useful to first learn modal logic. This looks like a good book if you just want to jump right into it. There are also other books on the amazon page just get the one that best fits your needs. I didn't find that many resources that looked great on temporal logic this, this, and this are some pdf's, but I think what would be best is to look at the Stanford Encyclopedia of Philosophy page for each of the topics you're interested in, read the articles, and look at the bibliographies for more information. Good Luck!