pccp

I am at my wits end trying to research this stuff and I feel like even more of a noob than before. I rode this bike 20 years ago in college and I tried it again recently and fell in love with it but I'd love to make it more functional. The first problem is that it has 52/42 front and 13/28 back, 12 speeds total and the gearing isn't enough for where I live. I'd like to add a triple front at the very least before I think about changing the wheel and adding a higher count rear set. Also can I change out the brake pads for a modern one? I found koolstop holders for 25 dollars a pair but I'm not sure I can put them on (the brakes are shimano 600) or if it's even worth when I can probably upgrade to a more modern dual pivot for not much more?

The biggest thing I wanna change though again is the gearing, if I add a triple front crankset 52/42/30 which I found on ebay for $67 with shipping, Sugino crown triple 130 74 bcd, would that work with the bb axle that's on there currently or would it need to be longer? Also would the front derailleur and rear derailleur need to be changed? The front derailleur that's on there now has a much wider range with the friction shifter than the double chainring needs and I know similar models of this bike back in the day had a triple front chainring.
I have more questions about front and rear derailleur compatibility for future upgrades but for now I mostly wanna know how to get a triple on the front for the least amount of money and I'm eyeballing that ebay sugino hard. If the derailleurs need to be changed though, I don't know what is compatible...