I'm considering getting a power washer for the purpose of cleaning engines, bays and car / engine undersides of built up dirt and oil and was wondering what the minimum PSI and flow rate required are to accomplish this task, as I don't want to buy a more expensive washer than I have to.
As DustinDavis mentioned in the comments, high pressure water is very easy to damage components with. The area underneath the hood is not built for high pressure water (or large amounts of water at all).
With most surfaces on a car, the proper way to clean them is by hand. You can purchase engine degreaser products at most auto parts stores. This product combined with hand brushing and a common garden hose with a spray nozzle set to 'flat' mode will be sufficient to clean the engine.
Things to keep in mind:
Remove the battery and cover the connectors before cleaning. Coating the engine with water exposes any damaged wiring to a large amount of water. Removing the battery can help protect the vehicle from damage.
Plug the air intake with a towel or other means. Hydrolocking will destroy your engine in a single turn of the starter and require replacing the engine. This is something you really want to avoid.
Don't force water into nooks and crannies. The water is for rinsing, not for blasting dirt off.
Let your vehicle dry with the hood open in a well ventillated garage or outside on a sunny day for multiple hours. Engines are best run when they are dry.
Avoid directly blasting hoses, wires and other cabling with water. They're not made to withstand that, so best to not force it on them.
Apply the degreasing product to all visible surfaces that are obviously covered in grease and grime. Use a brush to break up the grime and work the degreaser into the grease and grime. Use the water to rinse off the degreaser (not to rinse off the grease directly). If the gunk doesn't want to come off then it means you need to repeat the process to get that gunk off.
There's a reason you find all of these components sheltered away behind body panels and up off the road.