A possible first bike could be a Royal Enfield, which is basically a 1950s British 500 c.c. single cylinder design which was also built under license in India. Although the British Royal Enfield company went out of business many years ago, the Indians have continued to build pretty much the same motorcycle ever since!
They are now available in the USA and are basically still the same 1950s British designed bike. According to owners, you get the authentic 1950s experience when you own one of these (i.e. lots of periodic maintenance and unreliability).
They have different versions and some even have such “luxuries” as electric start, disk brakes and electronic ignition.
Larger engined (modern design) motorcycles with a retro appearance include:
Triumph Bonneville, T100 or SE
Kawasaki W650 (no longer in production)
Moto Guzzi V7 Classic or Cafe Classic
Ducati GT1000
Harley Davidson Sportster (since 1957 but getting a bit posey now)
Honda Shadow RS (Sportster lookalike)
Any number of cruisers (which are basically styled after old V-twin Harleys)