Okay, I'm a Plumbing Specialist at Lowe's. Here's what to try on your own.
Assuming you have a two compartment sink, fill the side with the disposal with hot water, hold down the stopper on in the other compartment and turn on the disposal. If this didn't clear the clog or caused a leak, we go on.
The trap is the U shaped thing at the lowest point under the sink. You can often remove it by hand. We hope it's made of PVC because 20 to 30 year old metal may crumble if you remove it with a pair of channel-locks or pipe wrench. Go ahead, this is easy to replace. If clearing the P-trap and re-installing it doesn't work, - go on.
Remove the trap again if you want to use and auger. It's even better if you can remove the pipe going to the wall so you have a straight shot for the auger. As you run the snake, rotate it and go back and forth, then feed more in until you feel you have reached the clog. Pull it out slowly. Once you get out whatever clogged the drain, have a family meeting about what teh garbage disposal cannot take.
Check the website for lowes.com for other ideas.