multistage amplifier design steps/process
What you are really asking is how to design a circuit to certain specs.  There is no single answer or method to that, and I don't really know how to teach it.  But, this ability is what makes the difference between a really good EE, and one that merely cranks thru numbers.

The first requirement is that you must have some intuitive understanding of what the various building blocks at your disposal do.  Then you creatively think of ways of connecting them together to achieve your goal.  You do this a lot with simple circuits, wonder why they don't work as expected, have the "aha" moment, fix it, try the next thing, repeat many times.

If you start doing this by 7th grade or so at the latest, you'll have some intuition by college when the professor assigns you to design a audio power amp.  That's one of the things you inevitably tried, and failed a few times, to design on your own.  By now, you've probably got a small working audio amp you've designed yourself.

Eventually you'll have run across common sub-circuits that perform functions that seem to come up often inside larger circuits.  These include emitter followers, common emitter amplifiers, push-pull output stages, etc.  You will have played with different ways of biasing single transistor stages.  You have analyzed other circuits and seen how experienced EEs have solved particular problems.  Some impressed you as "Hey, that's a cool idea", and you saved that idea away in your own brain.  Now it's your own cool way of addressing that particular problem.  Even better, you generalize the concept and use it in new ways when appropriate.

Those are all parts of real design.  That's a creative process, as apposed to rote cranking thru numbers to see what the currents and voltages are in a existing circuit.

However, none of the above tells you how to dig in and actually design the circuit.  As I said, I don't know how to teach that other then tell you to go back to 7th grade and start tinkering.

Since your early experiments you've learned a lot more about the theory, so now you understand why your previous attempts made noises on their own, blew up transistors, sounded really crappy, etc.  You've also been taught the eye-opening math of negative feedback, and actually understand things like power dissipation, how to calculate currents and voltages in a transistor circuit, and the like.  Now you're itching to design a better amp with all the new things you've learned.  You've even been assigned that as homework, so you can't wait to get started.  You'll show them the best amp a student ever came up with.  It doesn't get any better than this!

If at this point, several years into college, you have no idea where to start, you are on the wrong career path.  You will be able to brute-force or copy your way thru some problems, but it's too late for you to ever be really good at it.  Switch to marketing or run for elected office or something.