replaced tire, now wheel flexes side to side
There should be no flex.

It's possible you haven't reinstalled the wheel correctly; and the axle is sliding around in the dropouts. If that's not the case, I assume you're talking about lateral wobble when you hold the bike frame still and wiggle the wheel rim. That indicates hub bearings in dire need of replacement.

You say the wheel felt stiff before replacing the tyre; the chances are the tyre/rim was rubbing against the brake pads. In changing the tyre you most likely knocked things about a bit and nudged the brakes so that they no longer rub on the tyre/rim, exposing the play in the hub bearings that was already present.

Much less likely: If you're talking about a different kind of flex (i.e. the wheel rim is actually deforming under moderate force), you've probably cracked the rim while replacing the tyre.

It sounds like a trip to your LBS to me. Wheels should not flex in any of the ways described above, and a bike which hasn't had a tyre change in 29 years will almost certainly have been neglected in other ways too. You don't seem to have a great deal of experience, and most likely won't be able to spot potentially dangerous problems. Take it in and ask for a full service.