Depends whether you are scanning (1) film media such as negatives or slides, (2) original photographs or artwork, or (3) published photographs or artwork.
Category 1 is a special case requiring dedicated equipment to do properly, if you need more info let me know.
Category 2 is what I presume you mainly mean. If you are scanning a well-made print on glossy paper, 600dpi is about right. That is about the best resolution that the photograph paper itself can manage, so any higher than that doesn't add much information. If the photo is not very sharp and/or is printed on matte paper, you can probably go down to about 300dpi without losing much information. On an original print, you shouldn't have much trouble with grain unless the original photo itself is grainy. In that case you should treat it as Category 3.
Category 3 (published and grainy pictures) should be scanned to whatever resolution you need for your ultimate use of the image. Scanning to a higher resolution and then downsizing tends to exaggerate interference patterns ("moire") between the printing pattern (halftoning, etc.) or grain and your scanner's own pixel matrix. After scanning, you will likely need to employ a photoshop plugin or a standalone application -- there are several available, I use mainly Noiseware myself -- to reduce the moire effect. These are basically special blur filters that try to detect edges in the picture and leave them sharp while blurring noise in the solid areas.
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