I have an older Kenmore kitchen range which has been working fine until last night.

While ithe oven was on preheat I heard a phffft, saw a small flash like something had arced and then the oven quit heating. The top burners continue to work fine, but the oven does nothing whether turned to preheat or bake.

I'm guessing one of the elements burned out.

Is there a way to clearly check whether this is actually case, and that it is not some portion of the wiring or a switch that has gone out?

If it is a heating element, how difficult is it to replace and does anyone have tips on how to do it?

They occasionally burn out, and are no big deal to check and replace. With the power off, remove the screws holding the little plate the ends come out of. Pull it out to expose the connections. Check for continuity between the two terminals. If open, the element is bad. The elements are expensive.

I plan to spend what ever it takes to keep my old range going. I want nothing to do with the new digital controls ranges or having to run new power feed to accommodate a 4 wire plug.

To do simple checks like this you do need some tools. A test light, a meter, or a voltage detector might be the best place to start with. I came across the niftiest gadget for trouble shooting, a voltage detector. They work through the insulation of wires. There are several brands. I have a GB Instruments GVD-505A, less than $15 at Home Depot. Touch it to a hot wire, and the end glows red. Find the doodad that lights it on one side, and not the other, and you have the culprit. You do not have to open up housings and expose electrical contacts. You are looking at where your hand is, not where the meter is. Most people are capable of doing repairs and will get it going and not get hurt if they use a little sense. The voltage detector makes it even easier.