Some of the posts above about dpi are more relevant to slide or negative scans, but I see the Digitize Media website is pretty clear. I’ve usually scanned prints at 600dpi - gives a bit of wiggle room if you ever want to crop or enlarge, or even zoom in to read something!
Edit - you could scan a few prints at different dpi using your existing printer/scanner; it should help you decide the best option for professional scanning.