Good job, I remember that one, it was a nice looking bike. Ultegra if I recall, so the drivetrain should work very well.

For pedals, a bike of that 'calibre', common opinion would say that the appropriate pedal/shoe system to use would be something road-specific, like shimano spd-sl, or maybe speedplay.

Next most 'acceptable' would be a mountain pedal/shoe system (also commonly used/marketed as 'touring'), probably shimano spd, but maybe crank bros or speedplay frog. It's easier to find mountain/touring shoes that are not too hard to walk in. Also there are dual-sided pedals which are spd-clips on one side, and platforms on the other, so you can use cycling or street shoes.

Another option that would be looked on as unusual would be "clips", i.e. cages and straps. That would be a classic/vintage vibe that would look odd on that bike.

Another option would be just platforms and whatever shoes you want. Either bear traps (super cheap, will bite into soft shoes and lead to numbness) or platforms, preferably with 'pins' (distributes pressure evenly for soft shoes, pins bite in and give grip -- can also tear up shins)

But all the above measures of appropriateness are just opinion/fashion. In terms of functionality, any can be as good as any other, it's up to you to choose. Because of the cost of cycling-specific shoes, my (personal/subjective) recommendation is to start with pinned platforms and sneakers you already have, ride it for a while, and decide whether you want to spend $100-200 to upgrade to a pedal/shoe system.

Wellgo MG-1, MG-2, MG-3 are all very good pinned platform pedals, pretty lightweight, and can be had for a little over $20/pair/shipped, if you shop eBay carefully.

But of course, you've got a bike with pedals right now that you want to ride! So maybe the best solution is to buy shoes (and replacement cleats if you don't have the cleats that would have come with the pedals) to fit the existing pedals.