This table (in pdf) gives an overview of the agencies (and a few foundations) that are most active in funding academic research and what types of research they fund. The links take you to the funding page of the agency (which can sometimes to be difficult to find for agencies where funding research is not a major part of their mission), not just to the agency's home page.
When pursuing research funding from NSF, it's extremely important that you identify the program or programs within NSF that best fit(s) your research. A good place to start is the NSF website. This video shows how to the use the website to identify the program (and program director) that is most likely to be interested in funding your research. The next step will be to contact the program director to discuss whether your research fits their program.
NIH's RePORTER awards database has been around a while now, but many researchers aren't aware that NIH has been adding a number of nifty new capabilities that make this a powerful research tool that you can use to identify the IC, program and study section that best fits your research. This video shows how to do that.