Federal Grants
Most federal proposals must now be submitted through grants.gov, the web portal for government grants. The process is complex, but not impossible so long as you leave yourself plenty of time.
If this is your organization's first grants.gov application, begin the registration process right away--it can take several weeks. Save all PINs, passwords, log-ins etc. in a place where others in your organization can find them should you be run over by the proverbial bus.
Call the federal agency to which you're applying and ask for assistance. The program officers will be glad to help, but if you wait till a week before the due date they won't have time.
Submit the finished proposal several days before the due date. If you wait till the last day (or minute), the servers may clog up and your proposal may be deemed late even if you pressed "Send" at 4:59 p.m., EST. Submitting in plenty of time will also allow you to correct any technical problems.
Grants.gov is the portal for researching federal grants, but be sure to check the agency's website as well (HUD, NEA, NIH, Dept. of Education, etc.).
Other useful federal websites:
Washington State
Counties, cities and municipalities also make grants. Go to their home page and look under the subject area that interests you.