So this isn't a specific part compatibility question, but more of a general knowledge inquiry.

In general, if I'm looking to replace a part on a bike (shifters, derailleurs, brakes) to do an upgrade, what are general tips on seeing if parts are compatible with each other? Do manufacturing websites have a general list or do you go and ask them directly/ask about it here?

Rule 1 is stick to the same "speed".
Rule 2 is stick to MTB OR Road
Rule 3 is dont mix brands
Rule 4 is that there are exceptions to Rule 1, 2 and 3.
The Cyclists Touring Club forum have a campagtabilty chart for mixing Shimano and Campagnolo. The hybrid is often called Shimergo (useful for a search term).
There is more discussion and tips.

Generally you have to research the particular 2 components that you want to play together.

Shimano's web site has cross-line compatibility charts for their component groups. As MichaelW summarized:
-Match "speeds"
-Stay with the same manufacturer's components and within their type (road or MTB) and perhaps within the same "generation" like 7900 Dura Ace with 6700 Ultegra.

For example, you can use an Ultegra 9-speed rear derailleur with 9-speed Dura Ace shifters.

Do that and you are almost sure to have the new component work. Go outside these guidelines and it becomes a case-by-case recommendation and/or the need for some intermediary device (like a Jtek Shiftmate) to make them match.