I've got a strong preference for coffee brewed using an espresso machine (ie high pressure extraction pushed through a tiny hole), but I'm not really sure what the cause is - is it different levels of extracted compounds? Is there any pour-over methods (ie not pressurised) that are similar?
If you havent tried the classic pour overs with freshly ground specialty coffee, I would really recommend it. I used to be an espresso guy, but now I only drink filters.
What I care about is what the coffee tastes like. How do the different brewing methods differ in taste, and what causes them to differ?
Espresso and slow brew coffee are probably the furthest apart, and many people like one and not the other, it's a personal preference. But even if you like slow brew, there's not a single one that tastes better than all the others. That's a little like asking which wine tastes the best. Some wines taste objectively terrible, but after a certain point it comes down personal taste. Some people like darker vs lighter roasts, high vs mellow acidity, and on and on.
If you just need somewhere to get started, I would start with slow brew because it's much cheaper. A good grinder can be expensive, but there are inexpensive manual grinders that do a decent job. Watch some videos, buy some coffee, and see what you like.
You also have different variables in the taste itself. The sweetness, acidity, bitterness, earthy flavors etc. There is also the feel of the liquid, based on the amount of suspended soluble material and oils.
What causes the differences is primarily that different chemicals are present in the coffee, and more are created in the roasting process. Those chemicals all have slightly different rates in which they will become soluble based on temperature and pressure. A finer grind, higher temperature, or longer brew will all result in extracting more of basically everything, some of which might taste unpleasant to you.
Pour-over and french press are super lame, only good for occasional light drinking sessions.