Some salmon brake pads will certainly help your braking, but aluminum rims and new pads will help you a lot more. for a bike like this (not very valuable) and your usage (not a power user), a new, inexpensive machine-built wheelset bought online or from your LBS is more than good enough. You can have your LBS rebuild the current wheels with new spokes and rims, but the total cost will probably be about the same and machine-built is probably easier. With new wheels you will, however, have to also get a new cassette (rear gear cluster), unless you can find wheels with hubs that allow you to re-use your existing freewheel. Your bike is old enough that is has friction shifting, which means that if you get a new wheel with a cassette you can keep your existing drivetrain.
I am pretty sure there are machine-built 27" wheelsets with aluminum rims out there. If not, then yes, you would have to get 700c wheels. There is a limited tire selection in 27", but you can get, say, a Panaracer Pasela, which will do just fine. Your bike appears to have Dia-Compe centerpulls, which give you some room for adjusting the pads down, which you will need to do with 700c wheels since they are a slightly smaller diameter. If they are Dia-Compe 750s they will have enough adjustment room; if they are 610s they likely won't. Model number is indicated graphically somewhere. Sidepull brakes like the Sora, linked above, do not have enough reach for your application.