You've asked alot of questions that don't have simple answers. But first a question for you: Have you tried different tires on these rims? Are the rims tight with many different tires? Could there be a tire that's easier to change on these rims? What brand and model rims do you have? Some Campy rims are known to be a very tight fit with tires. How old of an "old steel bike" do you have? Changing rims because of tire fit is not something that's often done.
Switching wheels is easy. Anything with 700c rims and the correct hub over-lock-nut dimension should work, but if this bike has less than an 8-speed cassette, it may need a freewheel hub rather than a freehub. There are a number of potential variables you should have an LBS check for you. The shop I work at sells some low end replacement wheels for $45-$60 that are fine on hybrids and 80s-era steel bikes.
Rim are without spokes and hubs, and switching rims on a wheelset is non-trivial (to put it very very mildly). If you want to spend some hours studying wheelbuilding procedures and invest in a truing stand and dishing tool, and you're mechanically inclined and bold and brave, it's conceivable that you could do this yourself. But note that a poorly build wheel could be dangerous. A shop could rebuild the wheels for you with new rims on your old hubs, but most shops would rather sell you new wheels, and the new wheels would likely cost less.
Oil is good, grease is better.
2007 Peter Mooney w/S&S couplers/Terry Butterfly
1993 Bridgestone MB-3/Avocet O2 Air 40W
1980 Columbus Frame with 1970 Campy parts
1954 Raleigh 3-speed/Brooks B72