I am fascinated by mushrooms but don’t know much about them beside being able to broadly classify them. I know there’s a lot thrown around about how toxic some of them are, and not sure about how much is true and how much is hearsay, but I always err on the side of caution. You’re 100% right about the underside of a mushroom being an important identification feature, there are usually a few other clues but the most reliable way to tell gilled mushrooms versus boletes is to look if the underside has gills or pores.
I generally avoid touching them, using two different methods;
Depending on where your camera is situated on your phone, it’s not incredibly difficult to get a shot of the underside of a mushroom so long as the cap is a few inches off the ground. Bonus points as photos can look pretty artistic, too. Upping the exposure a bit helps since they’re almost always in shadow.
Otherwise I’ll usually take a stick an poke them over, but only if there appear to be several of the same species in an area. I’m always careful about being destructive for the sake of an observation, especially if it’s something I know very little about.
Good luck and have fun!