As indy hints, component swapping can be either a simple change-out or it can lead you down an expensive rat hole. It's probably worth finding a bike shop with knowledgeable mechanics to discuss your choices, if you don't know which way to go.
FWIW, my loaded touring bogey for gears is a low around 20 gear inches. I'd start out by trying a mountain compact double crank -- that's probably $100-150 (all cost guesses in US dollars!), something like a 24-38. If that works, great, you're done. But I'm not familiar with Shimano Claris, you might need a NOS Tiagra or 105 front derailer. Worst case, you need to change to bar-end shifters, which requires a lot of labor to recable and re-tape the bars. As long as you stay with 8 speeds, the chain and rear cassette should be OK.
OTOH, the Fuji's wheels are machine built, which work just fine for some light, and lightly loaded, people. If you don't fit those categories, or are just cautious, ask around for a wheel builder recommendation. You want someone to make sure the wheels are adequately tensioned and stress-relieved -- if the alleged expert looks blankly at you when you say that, keep looking! It's slightly tedious to do it yourself, but knowing how to do it will allow you to fix any problems that crop up on the road. Jobst Brandt's book, "The Bicycle Wheel," is an excellent reference, and there are a couple of web sites that have adequate instructions.
