There are thousands of places to find geospatial data on the web. Below are projects of the Cornell University Library and other organizations that collect, organize, and provide access to geospatial data.
Geolode is a collaborative catalog of geospatial open data websites around the world, searchable and browseable by location, topic, and other tags, so that searchers can quickly focus on the most relevant websites for their geodata needs. It was originally created by staff at Mann Library, and is maintained by a group of volunteers from several colleges and universities.
CUGIR, the Cornell University Geospatial Information Repository, provides free and open access to geospatial data for New York State, as well as worldwide geospatial data created by researchers at Cornell. CUGIR provides dataset descriptions and previews, and in addition to traditional data downloads, CUGIR also provides WMS and WFS web services for most datasets.
New York Historical Aerial Photos
Cornell IRIS has a collection of historical aerial photos covering nearly all of New York state from the 1930s to 1960s. Six counties have been digitized and are available online. A project is currently underway to digitize the complete collection photos of all counties.
New York State GIS Clearinghouse
data.ny.gov is New York's primary site for open government data.
When searching the site, trying limiting the results to "Maps" (under "view types" on the left), which will limit the results to geospatial datasets that can be directly used with GIS software. When downloading a dataset, click the "Export" button and look for a "shapefile", "KML", "GeoJSON", or other GIS data format. For many point datasets, you'll need to download as CSV, which is a tabular format that contains longitude and latitude coordinates.
Most of the other datasets on the site will be non-spatial tabular datasets (spreadsheets), but some of these datasets contain placenames or standard identifiers that can be used to join the data to GIS polygon layers (for example, county-based statistics).
Discover GIS Data NY
Discover GIS Data NY provides access to elevation and imagery data for New York state. The site allows you to preview and download data, including DEMs (digital elevation models) from a variety of statewide and local sources, LiDAR pointcloud data from various surveys, and aerial imagery for specific years (1990s to the present) from the the NYS statewide orthoimagery program. To get to the previews and downloads, see the "tools" icon in the top right of the map.
For the aerial imagery, it is also possible to link to a web service rather than downloading specific image tiles. For more details, see the NYS Orthoimagery Web Services webpage.
Olin Library Map Collection
Before there was GIS, there were maps. Olin Library has a extensive collection of over 300,000 paper maps, books and atlases, and data discs, representing information that often cannot be found online. Library staff can help you find and scan historic maps, or to access datasets on disc. The Map Room is located on the Lower Level of Olin Library.
OpenStreetMap is a great resource for roads data around the world, and for many regions it is also a great resource for other data too, like footpaths, rivers, building outlines, landcover, etc.
You can visit OpenStreetMap.org to preview the map and see what data is available for your area of interest. The data in OpenStreetMap is usually very high quality, but one thing to watch out for is that it is not necessarily complete. Roads are nearly complete globally, but other types of features may vary from place to place. For example, the absence of building outlines in one neighborhood does not necessarily mean that there are no buildings there. It may be that no one has mapped them yet. (Data in OpenStreetMap is mapped by volunteers around the world.)
There are various ways to get that data out for use in GIS. GeoFabrik generates extracts by country and states, updated nightly, and makes the data available for download as zipped shapefiles -- see the ".shp.zip" links here:
Once you unzip the download file, you'll see several different shapefiles, for different things like roads, buildings, landuse, etc.