You might find that a complete crankset is cheaper than chainrings, assuming 2 & 3 rings can even be fitted to your existing cranks. You will need chainrings/crankset, front deraileur, front shifter & cable, chain and unfortunately I suspect cassette and rear deraileur as nothing modern supports a 2x6 or 3x6 that I'm aware. (7 is the minimum.)

Actually, I would suggest that you buy a complete take-off driveline from eBay, Local Bike Shop, etc. If you don't get a complete package you will have to do some detective work to make sure you have compatible parts, and I highly suspect you'll be breaking the chain and adjusting the length by hand.

I'm new around here, but perhaps there are technical/mechanical discussion forums where you can find details about this. Your proposed change isn't that different from a mountainbiker who wants to go from 3x8 to 3x9 or 2x10, and I'm sure people do that.