just to be clear - the old three speeds are heavy - with fenders and chain guard at least 30 pounds. the upright position and the weight mean you will not be enjoying a Lance Armstrongesque ride. The British models will have odd threading and the Japanese or American models all use basic utility parts.
These are not good choices for someone who wants to go super fast. Though the nicer models track really nicely and give a good zippy ride when set up properly. They are very comfortable, with slack geometry and long wheelbases.
I always replaced the cranks and/or rear wheel sprockets on mine to lower the gearing but that's because I climb over the bridge to/from Queens every day. I replaced the sit up and beg handlebars with a cheap mtb bar and stem to get more aerodynamic (you can also just flip over the stock handle bars). I also removed the fenders and chain guard to lessen the weight because I ride in bike clothes and change at work, so I don't care how dirty I get while riding. My last Rudge weighed about 26-27 pounds after modification and was geared very comfortably for the bridge.
Three speeds in their stock form will give you a perfect all-around single speed when ridden 'direct drive', a slightly easier gear for when the road goes up a little (Manhattan is NOT entirely 'flat'), and a high gear so you can cruise at 20 mph or so on the Green belt path or in Central Park. Shifting is easy and simple.
The bikes being recommended here are all beautiful and if they mow your lawn , by all means go for it! If you just want something reliable, not flash - go with the three speed.
I'd say pay $50 for the lesser brands, maybe $100 for a top-line Raleigh or Schwinn in great shape. Factor in the cost of a good initial tune up if the shifting/braking is messy (usually they're fine), fresh tires (about $50/
pr.), and you'll probably invest no more than $150. Add maybe another $35 for a good Wald basket or a rack if you need one. After that you can forget it. Put a few drops of oil on the chain and in the hubs once or twice a year and they'll run forever.