I have just emailed the following to my first-year students.
Professor Sir Timothy Gowers, FRS, Fields Medal winner, Cambridge, is an incredibly successful research mathematician who, nevertheless, really cares about teaching mathematics to first-year undergraduates.
Although the syllabus they are following in Cambridge is a bit different to the one in Nottingham, there is enough in common that you should find what he says helpful.
See http://gowers.wordpress.com/2014/01/11/introduction-to-cambridge-ia-analysis-i-2014/ for his introduction to first-year analysis in Cambridge.
See http://gowers.wordpress.com/2014/02/03/how-to-work-out-proofs-in-analysis-i/ for some discussion of proofs. I like his immediate illustrative example of how understanding aids memory, which I quote here:
“Suppose I were to ask you to memorize the sequence 5432187654321. Would you have to learn a string of 13 symbols? No, because after studying the sequence you would see that it is just counting down from 5 and then counting down from 8. What you want is for your memory of a proof to be like that too: you just keep doing the obvious thing except that from time to time the next step isn’t obvious, so you need to remember it. Even then, the better you can understand why the non-obvious step was in fact sensible, the easier it will be to memorize it, and as you get more experienced you may find that steps that previously seemed clever and nonobvious start to seem like the natural thing to do.”