So, I have a 1981 Schwinn Sports Tourer that I inherited from my father
Would pictures help for those unfamiliar with the model?
This?
Anyway, you should probably post picture, because posting bike pictures is always a good idea regardless of context.
What sort of TLC?
I'd like to re-wrap the bars
If you want to keep the tape original-esque, it could be tricky. A lot of Schwinns from that era used Hunt Wilde plastic tape, which hasn't been made in decades. NOS Hunt Wilde tape still shows up in places like ebay, though.
An alternative shiny thin handlebar tape is Benotto Cellotape, which also shows up in places like ebay. It uses celluloid in its construction and has an iridescent appearance.
There's some modern reissue Benotto Cellotape available, although it's supposedly less iridescent and seems more like just a plastic tape.
RJ The Bike Guy
made a video showing how to wrap bars the same way that Schwinn usually did it from the factory on bikes like yours.
Another option which would still be period-correct is cotton cloth tape. It was standard for a very very long time. Here's a freshly-wrapped example from my stable:
You can wrap cotton cloth top-down like in the RJ video. When wrapping top-down, the end of the tape can be secured in place by the plug that goes into the handlebar end.
In my case I wrapped it bottom-up, and secured the tape
with a twine wrap. People usually use electrical tape to secure the end of a wrap of modern bar tape, but with cotton cloth tape it looks a lot better to tie it down with a string such as hemp twine.
But ultimately, wrap your bars however you like.
On my '79 Fuji, I currently have the bars wrapped with modern cork tape. It's thicker and softer than the original cotton cloth that came with the bike, and it looks different, but it hardly looks bad:
It's been years at this point since the last time I cleaned and oiled the chain and gears and since then I've ridden it maybe 10 times for less than a mile each ride since then 5 years ago.
Check the chain for "stretch." Chains don't really stretch per se, but the link-to-link distance grows as the pins wear down. An elongated chain will shift worse, and will wear the teeth on the gears faster.
There are tools for measuring chain stretch, but
you can do it with a ruler, and the ruler is probably the more accurate approach.
Too much elongation and the chain should be replaced. If there's
a lot of elongation, the old sprockets will most likely not mesh very well with a new chain anymore, and you might need to replace the freewheel and chainrings in addition to the chain.
You should probably lube at a more frequent interval even if the chain isn't dirty. When I use web lubes, I usually re-lube a chain that's been sitting around for a few weeks even if it's clean, because the lube tends to get dry and sticky over time.
Re-lubing with a wet lube involves drizzling a considerable amount of lube onto the chain (usually dripping over one spot while rotating the drivetrain backwards), waiting a few minutes, and the wiping off the excess with a rag. Unless the chain is severely gunked up, this tends to do a pretty reasonable job of both cleaning and lubricating the chain.
Tips on general maintenance
Depending on how much you care, it would be good practice to overhaul all of the bearings in the bike. Open them up, clean them out, repack them with fresh grease and new balls. Those bearings are the two wheel hubs, the headset, and the bottom bracket. It's hard to guess what specific tools you'd need without looking at the components, though.
Are the currently able to adjust the seatpost and stem, or are they seized?
Ideas for part repair or replacement?
Depends on what needs repair, if anything.
Definitely get new cables and housing for the brakes, if it's been a long time. I'd recommend getting quality housing, low-compression or compressionless, at least on the quality tier of Jagwire CGX.
You should probably get new cables for the shifters, possibly also new housing. If the currently shift cable housing is bare stainless steel coil, you've got a bit of a decision to make: that stuff creates a spongier shift feel than modern shifter housing, but it can also
look awesome.
Are you contemplating upgrades?
The catalog indicates that the bike came with chromed steel rims. Besides being heavy, these tend to make for very poor braking, especially in wet weather. Aluminum rims could be much lighter, would brake better, and would still match the look of the bike's silver components.
If your brake pads can be adjusted downward by 4mm, you could even get 700c wheels as the replacements, which would give you a better selection of tire choices (27" wheels haven't been used on new bikes for several decades, 700c is still the current road standard).
Relevant to possible wheel choice, though... is your freewheel 5-speed or 6-speed?