For any reasonably bright object, googling the catalog number has become very effective. Here's a sample...http://images.google.com/images?hl=en&sa=1&q=%22NGC+253
You do get some mundane results, but more astro images are online than we can see in a lifetime. Just remember to put quotes around your search term, as in "NGC 253". Otherwise you're searching for anything with NGC, OR anything with 253 in it.
Once you have a "classic" catalog number (NGC, Arp, MGC, M, etc), you have lots of options!
For galaxies, the best central reference is NED. It was mentioned in this thread -- you can get to it after Exploring an object on the SDSS page. But you can also feed it an object name directly:http://nedwww.ipac.caltech.edu/forms/byname.html
Near the bottom of the results page is a count of images, observations, etc.
The best astro catalog access is at Simbad:http://simbad.u-strasbg.fr/simbad/sim-fid
From there you can get more info than you'd believe, including links to articles mentioning each object. Among the search results you can get to the Aladin java app, which can query hundreds of catalogs and sky surveys and merge all the info together seamlessly.
On the other hand, I doubt there are many objects within the range of a 10" scope that have not been imaged at 10" quality. The Palomar Digital Survey comes to mind -- covering the whole sky in much higher detail than any backyard scope could do. Here's one way to get to it:http://skyview.gsfc.nasa.gov/