For Federal Funding Opportunities:
"In a study of 10,000 federal proposals, the only variable that was statistically significant in separating the funded and rejected proposals was pre-proposal contact with the funding source. Chances for success increase an estimated threefold when contact with the funding source takes place before the proposal is written." (Source: David G. Bauer, "How to Grants Manual," 4th edition, 1999; Chapter 10.)
For Non-Federal Funding Opportunities:
Network! Discuss your research with potential funders.
Eg; NSF and NIH regional conferences provide a great opportunity to learn about emerging trends and priority areas.