Oh hey, you're me a year ago! We purchased a home with a poorly installed & unmaintained sprinkler system, and I've spent many hours digging around the yard to learn about it. I also spent time asking the same question you're asking. Here's what I've learned:
Where do professionals normally begin to learn about them?
There are a variety of courses online or at local schools that will teach you about ideal configurations for irrigation systems. Some of these are linked in other answers, so I won't belabor them here. In most cases, these courses cover not only common layouts & types of hardware but also considerations like pump volume/pressure vs. pipe length, max valves (each valve corresponds to a zone) based on flow, and maybe even local codes to consider when it comes to crossing other water lines, buried electric/telco, etc. However, practically speaking, most of the professionals I know learned from other professionals. It's a really practical field & closer to a tradeskill than an academic area of study. Much like general contractors, many folks get a job installing sprinklers, learn on the job, and eventually spin off to work independently.
If you want to tap this sort of knowledge, a really good place to start is your local handyperson community. Set aside a small "education budget", and ask a few folks on something like Angi or Thumbtack to come "fix" something: maybe you want a new sprinkler head, or you need to reset a zone's depth. Hang out, ask questions, and learn from them. I've found folks in this field to be happy about sharing their craft, especially since it's often solo or small-team work. You can also pay them for their time to inspect your work and give advice. Especially if your irrigation is coming from a well system and not city lines, you likely don't need any formal inspections of the system and can work on it freely without permits (but always confirm this with your municipality first if you're not sure!).
How can I learn about how typical US residential sprinkler systems are supposed to work?
Here's the fun part: you don't. 😅 Sprinkler installation & maintenance techniques vary wildly depending on your soil type, climate, product availability, and code considerations. For example, here in Florida we can get away with much lower pump pressures due to the frequency of high-pressure groundwater sources. We also have some additional practical considerations to fixing pipes underground due to our sandy soil, which doesn't hold the pipes down very well over time. Installing on grades in mountain communities with lower-pressures? Or in very cold climates where freeze prevention is a major consideration? Do you need backflow prevention for public water line integration? All of these things factor in. As a newbie, your best bet is to talk to neighbors or local crews about what they have/use, and focus your study on that.
So how can you buy anything if you don't even know what you're supposed to buy?
I am not an expert. There may be areas where other considerations mean you need to have a cohesive installation using specific fittings, heads, etc. In my area, this isn't the case - all the systems I've worked on for others have been a hodge-podge of different brands, materials, and configurations. My home system is an ancient pump hooked up to the cheapest-possible "do it yourself" Orbit 3-valve kit with 2 functional zones (and 1 incomplete loop), spraying through a handful of Orbit, Rainbird, and Toro heads. These were all happily compatible with the standard 3/4" PVC fittings on the main lines, and I even mounted them on swing pipes to make setting the heights easier after have to re-dig the entire system because heavy rains caused it to float up! All of this work was checked by local professionals who said it's in line with what they see. In many cases, cohesive systems are a matter of professional preference and taste, or even just supplier contracts, rather than actual utility.
Irrigation's a fun place to learn about and work on because it's so flexible! You're not tied to particular kits, and you can experiment rather cheaply. For less than $100 USD, most folks can buy the heads, fittings, and extension piping to redo their system or add/extend a zone. I've learned a whole lot by just trying some differently-sized sprinkler heads, types of spray, etc. It's a very forgiving sort of plumbing work: if you do something wrong, you'll see the unexpected water flow (maybe a trickle, maybe a fountain!) and can adjust or reset the old fixtures accordingly. Take a trip to the hardware store and feel free to play with some of the components to understand them better: you'll see quickly that most heads & elbows, pipe, etc are interchangeable. Your biggest questions as an amateur exploring an existing system will probably come down more to "do I have enough flow to support this type of sprinkler head?", which you'll know quickly when the system activates and the water just dribbles out.
I think the TL;DR here is: take chances! Make mistakes! Get messy! Residential irrigation is relatively cheap, friendly for amateurs, and a low-stakes way to take more ownership of your homes' supporting systems. I'd discourage you from taking any sort of formal class until you have a little time in the dirt to understand your components better.