On washing machines, you will find it will be mostly belts, water level switches, solenoids(mechanical switches) and water pumps. Most are pretty easy to fix. On dryers it will be drum belts, thermostatic switches, door switches, heat elements if electric, and thermocouples if gas. Or the thing could be too full of lint to properly dry the clothes. Again mostly a matter of investigation to repair.
Refrigerators are prone to needing a defrost heater, or the coils cleaned. I have a 2010 model Ge range, and a 2010 model Whirlpool refrigerator, and both have had service calls under warranty in the last year, so I question the quality of the new stuff.
Use of the internet to find problems is a big help. In some cases you can email or phone a tech support person and get a quick fix.
What is not fun is paying some technician in a quirky little van to come and fix your appliance, as his service call is say $75.00 an hour, and he in essence would be doing the same thing you could do on the internet. The major difference in you and the tech is that the tech thinks about this stuff every day and you don't. Plus he reads (or should read) all the service bulletins put out on the crappy new appliances we have today. His parts are retail, and you can get the same parts wholesale at some outlets that you can find on the internet. FWIW: Lowe's has provided me with really good customer service on my new appliances that I referenced. I am a little leery on the new refrigerator and thinking of buying their in home extended warranty, with food loss reimbursement, later on. One call in this case would really equate to the cost of the extended warranty. In other cases, probably would not buy it.
Also be advised you can establish an account with an on line parts company like Maintenance USA just the same as an appliance company can. They usually only require a business name, a phone number and fax number to get you an account. Try something like the "ERE Apartments" or "Jacob's Shade Tree Appliance Repair".