I had a post doc research fellowship that started the summer after I graduated. I firmly believe it made a huge difference for my job prospects, for several reasons:
1) I went to a good, but not great, not Leiterrific grad school (ranked in my AOS, but just outside the top 20). My post doc gave me a modest pedigree.
2) My grad program wasn't so hot on mentoring. My post doc PI was awesome as a mentor, and took that role very seriously.
3) The post doc was pretty demanding about publications, so I went from 0 to 8+ during my two years. Add to that good mentoring about publishing, conferences, etc. That aspect of it was invaluable, and not something that I, realistically, could have done on my own.
4) Although mine was a research fellowship, in my second year I was asked to teach a grad course as a sabbatical replacement, which enhanced my teaching portfolio (which had previously been a lot of undergrad courses).
5) I had really good, multidisciplinary networking opportunities (funding for conferences, lots of interesting people at the university, etc.), and had excellent colleagues who are now lasting friends and collaborators. Of all the jobs I've had, my post doc was the Best. Job. Ever.
On the down side, the post doc meant packing up my family and relocating for two years, and then doing it all over again when the post doc ended. The payoff was that I did dramatically better in my final year on the market than in my first two years, and I landed a TT job, and I think that's largely attributable to my post doc experience. An unforeseen side effect of the post doc: I always saw myself teaching at a SLAC, but I never had a single interview with a SLAC. Only research schools seemed to be interested, and I suspect that was a consequence of having had a research fellowship. I can't speak to how teaching fellowships might affect one's profile.
Needless to say, I endorse post docs as a career enhancement, but also as a paying job that serves as a useful bridge between grad school and a permanent job. Some of the postings I've seen this year actually pay very well (one in Australia was over $90K, one at a state school in the US was $65K -- those are both well above starting salaries for assistant profs). It's possible to go from one post doc to another, although some have time limits (5 years post degree is typical).
Applying for research fellowships is not very different than applying for a TT job. The dossier is typically pretty similar, although you may be asked to specifically address how your research will be furthered by the faculty at the host institution. You may also be required to have a research agenda that's defined by the fellowship or the funding grant. References, writing sample, CV, cover letter.
If you have questions about applying, or about post docs in general, ask 'em. If you have advice about post docs -- post it.
~zombie
Applying for research fellowships is not very different than applying for a TT job. The dossier is typically pretty similar, although you may be asked to specifically address how your research will be furthered by the faculty at the host institution. You may also be required to have a research agenda that's defined by the fellowship or the funding grant. References, writing sample, CV, cover letter.
If you have questions about applying, or about post docs in general, ask 'em. If you have advice about post docs -- post it.
~zombie