How do I preserve my theme customizations when I upgrade?

Hello,
Warning: I am a newbie 😉

I am maintaining a 2.5.1 blog, which is the website for a non-profit animal rescue. I’d like to develop a new (2.8.4) one with a sleek Photoshop layout and customize the heck out of it to replace it. I haven’t upgraded yet because I’m afraid of losing the few customizations I’ve already made to the first one. I’ve read in the CODEX that you need to back up everything (obviously) before upgrading to avoid theme files being overwritten with the upgrade, but then what?
Let’s say I did a lot of custom stuff to my current sidebar.php file, such as showing only specified pages in a determined order. I can back that up before upgrading, but would the new sidebar.php file have code I should be putting into my old file? It would make sense that upgrades might bring changes to theme files, otherwise why would theme files be involved in the upgrade process? My point is that I don’t want to be doing extensive file comparisons for each file in my theme…
I would like to stay within the most recent versions of WP as they become available but I want it to be as risk-free as possible to avoid breaking my site and custom work.
Secondly, does anyone have suggestions for how I can develop my new theme, using the same database that feeds the “live” site, on the same server & host? I’m not sure how I can make a dual dev/production environment for the same website that allows me to test the work and do a clean swap when the new one is ready. Mostly I’m thinking that links in the new will break in that transition.
No advice you can give is too lengthy; please be as detailed as you can 🙂
TIA,
VM