If you're just weighlifting or doing the elliptical, go with something that gives off a warm Americana vibe. Something like Vans plimsolls or Chuck Taylors are great and flat-bottomed for lifting weights, and pack flat since they're so minimal in contruction. If you want something more fleshed out a retro tennis or indoor soccer shoe would suffice, like Adidas Sambas, etc. I'm partial to these retro Adidas, though almost any of their Originals models, and similar ones from plenty of brands, would work:
If you want a runner that will suffice you can go with flyknits if they're your style; some retro runners in the style of Asics/New Balance/Sauconys/DIadora would get you through light treadmill work if you are into the chunky retro runner look.