I carry an Army M3 medic bag stuffed under the rear seat of the truck, with the carry webbing prominent to get it out via the strap.
It used to be set up for GSWs sorta per doctrine. These days I stock it with things more pertinent to daily events. I try to make it happen to bring it in the house to check status of items twice a year. Doing that, helps to maintain familiarity with its contents and where things are. Helps also with getting rid of dried-out prep pads and other things that come packed with moisture as part of their use.
Having what used to be called a Field Surgical Kit in that bag is a nice thing. That kit is an OD canvas roll, containing straight and curved forceps, a decent scissors, tweezers, scalpel, and places to put needles and thread and extra scalpel blades. The FSK has gotten more through the years than have the other contents of the M3 bag.
I got my first aid indoctrination at Ft. Sam Houston in a 10-week Combat Medical Specialist course a generation or so ago. Some things have changed since then, and some have not.