feels like a basic thing but I'm not sure how to get a series of points from a single emitted particle's position over time and create a curve from the points.

Add sop, by attribute, ‘id’.

http://www.tokeru.com/cgwiki/index.php?title=HoudiniDops#Pop_trails [www.tokeru.com]

You don't necessarily have to emit additional particles, use the Trail SOP, Connect as Polygons (turn off Closed Rows), set the Trail Length to the lifespan of your particles. The Trail Increment is how many samples along the particle path