In my first year of a research postdoc in math, I think I may want a teaching-only career. How should I prepare for the job search?
Not as a complete answer, but as information the questioner and others might find useful:

At an R1 U.S. university, for example, it would be difficult for a postdoc to make a transition to a longer-term lecturer or even adjunct... if only because those positions are spoken-for, and quite competitive in a certain localized sense, which happens (in my observation here in Minneapolis) to not be entirely bad, insofar as there is tacit (which is bad) recognition (which is good) of seniority and experience and such things.

A few faculty may be "outraged" that you "betray" the "research Grail" by deciding as you have, but you have to write that off. Most faculty will be very understanding that you've "achieved clarity", for whatever reason. They will also understand that you need to be able to document teaching and pedagogy and mentoring. NB, it's not just "teaching", but a larger thing, "pedagogy", and a somewhat-different thing, "mentoring". Senior-projects, participation in REUs, and similar, are relevant.

And, somewhat nonsensically, an unfortunate fraction of the up-scale four-year colleges have created the concept that "undergrads do research", so you should be careful to not over-promote any potential disaffection with (actual?) research in your job apps. That is, the spin in those cases would have to be that you've had some epiphany about preaching to the masses about . You'll not want to play that stunt for serious undergrad teaching places, though, because they'll probably think you don't understand (the) reality. :)

So, yes, actively discuss with your faculty there, and actively arrange to have people (faculty) have some acquaintance with the broader range of your activities. NB, "not just teaching", but pedagogy, mentoring, ... maybe outreach... The sane people will be understanding, if maybe not very interested personally, and some should be interested enough to help you document your efforts (presuming you make them!) in the directions you want to claim you care about.