Here are some questions to ask when you see a funding opportunity of interest:
How much money? Does the funding amount match your needs?
Am I eligible? Are you ready and do you meet the requirements? Read the guidelines carefully, then read them again. The guidelines will tell you about: submission deadlines, eligibility, requirements, proposal format (margins, spacing, maximum number of pages or words, etc.), evaluation process and criteria, award amount(s), whom to contact, etc.
Is there a fit? Do my research interests map to the sponsor's interests? Look for a list of projects previously funded through this specific grant program on the agency website; an annual report may also be available which will help you understand their mission and priorities. Be honest whether the funding announcement matches your research project.
Do I have time? Do you have time to prepare a competitive proprosal? Do not rush! Expect to wait 6-12 months before you receive funding. Factor this into your timeline. Note if the agency's website mentions # of applications and the # of awardees each year. Receiving a grant from Fulbright, NSF, or Spencer is prestigious, but competition for these grants is fierce.