Your best bet is to start with people you have collaborated with on your current research. Co-authors, consortia, at your and other institutions. Your mentor(s) also should know people who are looking for more help, so they will also be able to put you in touch with other professors. Aside from that, every institution will post open post-doc positions on their websites and journals will also advertise positions open. Those advertised positions will be far more competitive, however.
Generally speaking, most academic post-docs are geared towards others who want to continue in academic or research institution (ACS, NIH, Sloan, Mayo, Venter, etc.) jobs because you'll be doing that kind of work. Industry will post their own post-doc positions and are generally advertised on their own websites, and occasionally in journals.
They'll be competitive according to how known the researcher is. The more known the researcher, the more competitive the position will be. Kind of how programs at top notch schools is in the single percentages. But remember, that anyone looking at post-doc job is likely a very good candidate and was very successful in grad school.
