Welcome to the forum, arron!
Your American Flyer locomotives are known as 'S' gauge trains. The 283 dates from about 1954-1957, and the 290 dates from about 1949-1951. The Lionel 1073 transformer was manufactured from 1961-1966.
Getting continuity between the 2 prongs of the AC plug is normal. You are reading the resistance of the primary winding of the transformer. If you did not get a reading, there would be a problem, either a bad wire, or a bad transformer.
Was there anything connected to the transformer when you plugged it in and the circuit breaker tripped? If there was nothing connected, then there is something inside the transformer shorting out the secondary winding, or the circuit breaker could possibly be bad. Do not worry about life or property, as the circuit breaker is protecting both. Unplug the transformer, and remove the 4 screws holding the cover in place. Look inside to familiarize yourself with the inside of the transformer. Look to see that all solder connections are good, and that there is nothing loose inside. Here is a page from the service manual, to show you what it looks like inside.
http://pictures.olsenstoy.com/cd/ho%5Cd1043a.pdf Reassemble the transformer and plug it in again with nothing connected to it. If the circuit breaker trips again, the transformer needs service. If it needs service, your best bet would be to replace it. There are many good, used transformers for sale at your local train dealer and on the internet. You will need an operating transformer to test your locomotives.
With an operating transformer, inspect the track and make sure it is clean and free of rust. Clean any light rust from the track, including the end pins, and trains with a ScotchBrite pad. Wipe off any dirt, grime and oil from the track and trains with naphtha, available at your hardware store. Assemble the track into a basic circle or oval, and attach the lockon to the track. Connect the wires from the transformer to the track, put a locomotive on the track, plug in the transformer, and turn on the power. The locomotive should move. If not, turn the power off and on again. Listen for a buzz from the tender. If the locomotive does not move, look under the tender and you will find a small lever protruding down. Move the lever to the opposite direction and try the locomotive again. If this does not work, and the headlight illuminates, then the loco needs service, probably the reverse unit in the tender. Try this on both locomotives. If both locomotives run, then they will need basic service to get them to run their best. That is for another day. First you need a working transformer.
Here are links for the exploded views of your locomotives. Look at them and familiarize yourself with the insides of your locomotives and tenders. The 283 is here,
http://www.hobbysurplus.com/xviews/280spacHSS.asp and the 290 is here.
http://www.hobbysurplus.com/xviews/290slcHSS.asp Keep us updated here with your findings and progress.
Larry