Invisible gloves work great but are a limited level of protection. Everyone should pick up a tube for a trial run.
A box of latex, nylon, nitrile.......gloves should be in every toolbox. Not only do they keep your hands reasonable clean, when you remove them, you won't leave grease/dirty finger prints all over the vehicle you're servicing. They do damage easily but act like a 2nd layer of skin. And, if you have sweaty hands, you'll never get the 2nd pair on. Invest in some climbing chalk or talcum powder. Nothing like smelling like baby powder when wrenching on someones car.
Mechanic gloves are for those with weak knuckles and grips. If your lil' lady pays for your manicures, then consider them mandatory. They'll also protect against brute force incidents and the occasional brush up against something hot. Didn't need them 20 years ago, don't need 'em today. Toughen up!
Any of the pumice type hand cleaners should scrub your hands clean. Invest in a hand/nail cleaning brush. If you run out of hand cleaners, dishwashing/laundry soaps, along with the bio spray cleaners that are grean/purple/yellow/orange have no problem with removing dirt/grease/....from your hands.
For normal soaps around the sinks/tubs/showers, stick with the vegetable oil based hippie soaps. They seem to work much better then cheap 'popular' brands soaps.
Start manually washing dishes, along with regular visits to the beach, pool, lake...... Extra time, with the hands in water, will clean 'em up and remove the smell of auto chemicals.