Step 1: Peruse the NSF REU webpage.
Get a rough idea of what’s out there. NSF is the single most comprehensive site of its kind for STEM opportunities for undergraduates. You will probably go back to it after Step 2.
National Science Foundation
Step 2: Discuss your research interests with faculty members.
Your professors and other mentors in your discipline are your most valuable resources.
Faculty membes can save you a considerable amount of time by helping you articulate your interests and focus in an area of research that is relevant and available to undergraduates. They will be able to discuss the relative merits of various programs and funding opportunities. They can point you in the direction of competitive programs, or steer you into less formal, but more available opportunities. They also know you and can help direct you to an REU that will develop your interests and provide skill sets you are ready to acquire.
Step 3: Research REUs including those supported by an organization.
Look again at the NSF website in Step 1. Also check sites organized by professional associations or discipline-based institutions which provide searchable information on REUs available at many universities by category, region, and level of education. Examples of some good resources of this kind are:
Institute of Broadening Participation: Pathways to Science
American Mathematical Society
American Psychological Association
Step 4: Go to specific college and university REU pages.
A Google search for REUs by discipline will yield a mass of opportunities worth combing through. This is the least organized and most time-consuming process but may well overturn a hidden gem that appeals to you. If you have a graduate school in mind, its page should be at the top of your list.