It has become extremely easy to search for previously completed observations in recent years. The following catalogs allow you to search by various criteria and locate data for objects you may be interested in.
Simbad - Contains data for any observed object outside the solar system. Simply input the object name or coordinates and a search radius. It returns a list of objects that match you search criteria, papers about that object, and data for that object.
NED - The NASA/IPAC Extragalactic Database contains information about nearly every extragalactic source observed to date. Here you can enter a galaxy name or coordinates, and the database returns a list of objects that fit that criteria. If you select an object, you can also get a list of papers that discuss that object. NED also contains tools such as various velocity and distance calculators, as well as the NED knowledgebase (also known as "Level 5"). Level 5 has many review papers that focus on overviews of different topics and results, which can be helpful when you are new to a subject area.
VizieR - This can be used to search for entire catalogs rather than just single objects. You can put in a survey name, survey author, or even the telescope that was used and it will return all catalogs that have been made that match your criteria.
Many large surveys also host their own databases where you can download all or part of the catalogs created from their surveys. One of the largest is that of the Sloan Digital Sky Survey (SDSS), which can be found on sdss.org and contains a convenient online search tool so that you only have to download the data you need. Other sites for smaller surveys have catalogs available to be downloaded.
CADC - The Canadian Astronomy Data Centre hosts data taken with telescopes that Canada have some share in. Similarly to the other databases, you can search by coordinates or object name. You can also search for data based on the telescope that was used to take it.