What resolution should I scan my prints with?

Just curious, what resolution is it best to scan prints at? I don't usually go higher than 300 since it slows the system down quite a bit to go higher, but I've been scanning some of these horse show pictures in at 600, since I hope to print some of them, and I've noticed the improvement is even more significant than I expected.

This really depends on what you're going to do with them. I personally scan at 300-600 and then change it to suit the application. For the web, I don't bother with anything over 200. The file size stays small and editing tools have more effect, making it a little faster to get postable results. For printing, I go with 600, since my home printer does best at that rez. If I want to burn up a lot of ink, I can go for 1200, but no one can tell if I printed at that rez anyway. At the shop we scan all prints at 600, and all slides/film at 4000, but this also depends on what the customer wants done. We don't print higher than about 700-1000, since, again, no one can tell except us anyway.
I would scan at 300-600 and then change it if you need to or just rescan at a higher setting for high-rez prints. The recent horse pics you've posted were just fine (but could use some unsharp masking to get them a little crisper).
Hope this helps!