I still keep my old Schwinn around... now it's a beater/errand/commute bike, but it always makes me happy to just take it out and wear normal clothes and shoes
Mine is a little older, and was possibly a little bit more high-range for it's time in their line-up.
Those are, BTW, stem shifters, not downtube shifters, and will actually be possibly a little easier to get used to for you. Use this, you'll build all the strength you'll need, and you'll appreciate a new ride all the more when you can get one eventually. Upgrading is going to be difficult and pointless in the long run, in addition to being expensive.
Shoes, seat and pedals would be a worthwile investment, you can always transfer them over to whichever new bike you wind up getting if you do so eventually. Shifters are hundreds of dollars (potentially) and would require replacing a good portion of the drivetrain (which=not worth the $) Civilians like us don't get a discount on components the way the manufacturers do... so we're pretty much S.O.L. in these situations.
I've had similar tires and wasn't crazy about the way they felt. If it rides okay, keep 'em, otherwise you can get continental Gatorskins in a 27" tire, which will take a very high pressure, and have some kevlar for flat protection. They are the most modern tire I've found for 27" rims for my antique.