We have just moved into a new house (built in 1950). When we turn on our garbage disposal, the drains on both sides of the sink shoot out really disgusting water with a ton of food, etc. in it. It truly shoots straight up into the air and pretty much gets on everything in the kitchen! Is this a disposal issue? Also, when we do laundry, the kitchen sink fills with water on both sides - soapy laundry water. The sink does not seem clogged with food, because although the disposal shoots everything out, the water drains. Is there a chance that something could be installed incorrectly? Please help! I need to fix this, and if you have any suggestions for me on where to start, what it may be, or how to fix it, I will be forever grateful! Thanks so much!

Sounds like you have a blockage downstream frrom the kitchen sink that needs to be snaked and cleared plus a possible blockage in the horizontal drain in the wall. Your disposals table acts like centrifugal pump when it spins forcing the water down the drain. If the drain's blocked it will pump the discharge up out of both of the sinks tubs.
The solution would be to snake the drain from the trap to the kitchen vent stack and then go on the roof and snake from the kitchen roof vent.
Good luck, Tom

Thank you so much for your help! I honestly didn't know where to begin! Do you know of any websites I can go to for a visual reference or diagram?

What is it you wish to know? You snake the lateral drain in the wall by removing the "J" bend from the trap and snaking through the wall stubout.
You snake the main drain line under the slab by snaking from the kitchen roof vent. You put out enough cable to reach the base and 20 feet more. Test by cycling the washer. good luck, Tom