I worked for years doing appliance installation, sales, and minor repair... Washers and dryers are really easy to fix but 9/10 the difficulty comes from finding parts. If you are at least a tiny bit handy I bet your pump is clogged. I once found an entire pair of panties sucked up into one of those things it was crazy. If its already broke, you cant hurt it by just taking it apart... be warned, remove all the water first with a shop vac... you start disconnecting stuff down below your going to get a floor full of grey water... You could have thrown a belt off the transmission/motor, etc etc... Thats about the best I can do for you online...
Now if you are going to buy a new washer and dryer, check your local brick and mortar, ma and pa stores first... These can be better sometimes at giving you good deals and they typically will handle warranty issues for you and have they own service technicians that wont charge you an arm and a leg should things go wrong... Otherwise you know your big box stores and you can get a good price if you pick up and install yourself.
Now, if you want a brand recommendation....
Stay away from GE, Kenmore, and maytag.. This is my 2 cents because I have pulled a lot of these out of homes... the older ones like maytag were tanks... anything newer... well... yah...
Good brands are, Whirlpool, LG, Fridgidaire, And I have had some good luck with Bosch, and Samsung. But the first three would be my recommendation. Fridgidaire will be more basic, cheaper, and heavy duty, easy to work on, and thats what I have in my home. Whirlpool will be your next higher end brand right next to LG... Fancier, prettier, more features, most have a computer in them so... you have to know what youre doing to work on them.
NEXT would be go front load or go home... these days it just makes sense, and some can be found for around the same price as top loads... They use less water, clean better, use less soap, use less energy, can do bigger loads, and are better on clothes. They are by far a more efficient machine.
Good luck out there, and yah, when something breaks like this it is prettier sickening, but you will get it straightened out!