I think I phrased the summary/title poorly but essentially, I'd like to know if all drywall mud coats need to use the same type of joint compound for all coats or if I can transition to quick drying mud in later coats.
Here's why I'm asking -- I'd like to learn the basics without being rushed. Assuming that goes well, I'd like to switch to quick-drying mud so I can complete my current drywall project faster.
You certainly can put different types of mud on top of each other without any problems. Generally it's done the opposite way though - using the quick drying mud for the first/second coats and the pre-mix for the final coats.
As for sandability, pre-mix is going to be the easiest to sand. The quick drying mud can still be sanded without much trouble though, depending on the type you use. Typically you'll see 90 minute, 45 minute & 20 minute varieties. The faster drying ones sand harder than the slower drying ones. The 90 minute stuff is still very sandable, 45 is still doable, but I wouldn't want to sand the 20 minute stuff.
Also, keep in mind that the fast drying mud has a limited work time, and you can't just add more water to it to soften it up like you can with the pre-mix. If you mix some up, and it starts to set, you just have to throw it away and mix a new batch. This usually isn't an issue with the first couple coats since you go through mud pretty quickly, but the last couple coats are much thinner so you aren't using nearly as much mud.
One other tip - I usually keep two buckets of water around when I'm using dry mix mud. One with clean water that I use to mix the mud, and another with gray water that I use to clean my tools between batches. If you use the gray water to mix your mud, the mud will set faster than if you use clean water. Works out nice if you are using 90 minute but want it to set faster than 90 minutes.
In addition to the things @Steve Jackson mentioned in his comment, the quick-drying mud (called setting compound) is also much harder to sand than the slower drying-type compound, which makes it not such a great choice for your final coats where you're going to be doing the most sanding. Pros use it because they're able to apply it so well that it requires little or no sanding, and it saves them a ton of time. In my opinion, non-pros (including myself) do best to reserve it for those couple of joints that are wider than they should be and call for stronger compound that doesn't shrink.
Also, with drying compound the later coats will take less time to dry just because they're thinner, so it's not going to delay your project as much as you might think after waiting for that first thicker coat to dry.
Quick setting mud dries even quicker if you use hot water!! the final coat should be regular mud and it can be helped out with a heat gun or a hair dryer.
