This guy is a shaving guru; he has dozens of YouTube videos about how to “shave like your grandpa”. After the initial investment of the shaving mug, soap, and brush, the lathering is much cheaper than buying the canned stuff.
What I’ve found works best: like everyone has said, moisture is key. I soak a washcloth in screaming-hot water, DON’T wring it out because I want it wet, fold it once diagonally, and press it against my face in the “robbing a train” position. Rub it around a bit and hold it there until it cools, then heat it up and do it again, and again, and again, four or five times, or even more. The heat and moisture soften the whiskers and prepare your face for the blade.
Before putting the lather on, I splash my face generously with hot water one last time, as a base lubricating layer under the foam. Some folks use shave oil at this stage; I wouldn’t recommend that for a teenager whose skin is probably plenty oily already. Lather on, and shave, with or against the grain, whatever you find works best for you. Rinse the blade regularly while shaving; a clogged razor will stutter across your skin and cut you. Go for the big areas first, then touch up the small areas last (clean up sideburns and such). Rinse everything off with more hot water, clean the razor well so it’s ready for next time.
Last, aftershave. Again, it’s gonna depend on what works for him. If his skin is oily, an alcohol-based aftershave may work fine; I sometimes use Royall Lyme, smells great. Most days I prefer a lotion-type aftershave; Nivea’s Post-Shave Balm is a favorite, not too thick or greasy. Try different things until you pick a favorite.