If you have steel rims, upgrading to aluminum ones would be a nice upgrade, especially as far as braking is concerned.

If your original rims are 27", you've got a few things to think about...

27" rims haven't been offered on new bikes since the '80s, so new ones aren't plentiful. They are still available if you look for them, though. You can pick up a 27" wheelset from a place like Niagara Cycle for a little over $100. Used sets are an option, too. (I happen to have a spare set in good shape. I'll probably never use them and would be happy to ship them to a buyer who'd put them to use.)

Switching from 27" to the more ubiquitous (and slightly smaller) 700c size is possible and increases your replacement options, but it requires that your brakes be able to reach 4 mm further. If you can't adjust your pads down 4 mm, you'd need to replace your brake calipers to accommodate 700c rims.

This could get more complicated. If you want indexed shifting (one click per gear), you'll need shifters, derailleurs, and a cassette or freewheel that are designed to work together. And a hub on the rear wheel that will take said cassette or freewheel. And a clamp that'll fit your frame and the shifters if you want downtube shifters. And frame dropouts that are spaced to 126 or 130 mm. There are a number of variables here, and it'll take a bit of knowledge and/or experience to sort it out. It's definitely doable, but might be cost prohibitive if you don't have a convenient source of used parts.

Depends a lot on the particular brake, the setup, and the type and condition of pads. There are decent brakes of both types.