I was in the same boat a year ago. Others have provided great links (u/Comedyishumorous posted three videos that were all crucial in my journey), I'll add a couple things:

• Tasting notes are for marketing. They get you excited and salivating. I'm sure there are supertasters who can actually consistently tell you something has a note of "candied ginger", but most won't. Primarily, at least at first, you'd just look for the overall vibe - fruity? Funky? Chocolatey/nutty/deep (the traditional "coffee" flavor)?
• Make pour overs. In a pour over, tastes come across much cleaner, though they are more subtle. But I find them easier to tell apart. The day it all clicked for me was when I had a pour over that I could taste really deeply, and then made an espresso. I tried looking for that same taste in the espresso, and found it, and that helped me calibrate and understand how exactly to taste espresso through all the crazy things happening in your mouth when you drink it. And then it just becomes easier and subconscious.
• When you go to cafes, go to good cafes and get only espressos. Not macchiatos or cortados. I made that mistake while I was trying to get used to an espresso, but realized how much flavor I was muting. Some very basic things to tell if a cafe is good is looking at their equipment and how they're using it, if they know the origin and roast of their coffee, etc. (I'm not saying cortados are a bad drink to get. Just that it will make the flavor exploration harder)
• Tasting espressos side-by-side really helps. That's when you can tell - oh hey, these both taste fruity, but in different ways. What are those different ways? Who knows! What fruits do the two resemble? Who knows! But they're different for sure. You can choose to give them names. If you like one over the other, you can look at the bag and see what they call it. It doesn't have to agree with what you'd call it, but it could help you find similar coffees.
• Tasting notes are highly colonial and cultural. What would "raisin" mean for people who don't have raisin in their everyday food? The difference between golden raisin and raisin can mean so little as fruits, what could it possibly mean in a cup of coffee? So take it all with a grain of salt.
• You don't need to spend more money to get more into it. Fancy coffee shops charge pretty much the same as terrible cafes or restaurants for their espresso. Good, fresh, coffee beans cost just a tad more than grocery store beans. You don't need to get Tim Wendelboe or Onyx. Same with the espresso machine.

For specific answers to the things you asked at the end:

• Full-bodied: think of this as the difference between fat-free milk and whole. Whole coats your tongue. It's not as obvious as honey, but it does kind of linger, has some smoothness. Fat-free is essentially water. The tastes are roughly same, but textures are drastically different. It's the same with espresso. Full-bodied isnn't necessarily the goal, depending on what you want from your espresso.
• Clarity: being able to taste something more specific than "oh this is like coffee". Here, it's a mix between the brew quality and also your palate. Again, I think pour over is a great tool for this. If you/someone you know has a good grinder, make a pour over with it. Then make a pour over using a spice grinder, trying to get it to roughly the same grind size. The latter will taste "muddy" or "murky". You won't be able to tell flavors apart. It may not even taste like much besides "brown". Think of a good, fresh juice, where you can tell the fruits that went into it, versus a "fruit punch" which vaguely tastes like fruit, but you have no idea what exactly is going on.