I know people can sometimes get a little touchy when it comes to the hobby of divination, so I'm going to answer your question, but preface it by saying you may not like it because I'm approaching it from a non-mystical historical context, and that while it comes from the perspective of someone who's done a lot of occult research and firsthand experimentation, they're also the words of a skeptic and a charlatan. So it's an accurate answer for me, but that may not be the case for you and I mean no personal disrespect if anything I say comes across as dismissive of what you may or may not believe. Now that my disclaimer is out of the way:
Truth is, it's always going to be clunky. Tarot cards were designed to be a game, and at some point it was decided “oh, and they can reveal secrets/divine the future and past/grant revelations too, sure, why not?” Since the mystical properties are arbitrarily assigned, it's really just a matter of memorization. It's very similar to if you had a standard deck of modern playing cards and decided you wanted to use them to read fortunes: that isn't what they're for, so you'd need to figure out what each card does in this new context and then memorize it. So if you want to get comfortable with it it's really a matter of practicing and remembering what each card means, as well as getting your inverses right if you use a spread that does inverses. Just like anything else: practice, practice, practice! You can actually find really cheap tarot for beginners at some Dollar Stores (or stores like them) that come with brief descriptions printed on the cards themselves. They're not the prettiest decks, but if you're looking to improve your technique they're pretty handy. Just don't use 'em to do a reading, because they're thoroughly unimpressive.

It can also be helpful if you create your own spread, which can be cool to onlookers because it's new and different. I used to run an Elder Sign spread with my Cthulhu Mythos deck; it was thematically appropriate, and because of the nature of the deck and spread I would put a bit of a darker twist on how I would read: here are the signs of coming misfortunes and evils in various aspects of your life so you can be prepared for them. If you're reading for other people, you'll also want to work on your presentation; if you work with a subtle tone, you can sort of “guide” your audience into really believing what they're seeing. I was never really good at that part, but for someone who really wants to floor people it's essential.

I fell out of doing tarot after the Cthulhu deck went missing so it's been years since I've done it (over ten…hoo, I'm gettin' old!), but hopefully my advice was somewhat helpful!

Edit: Also, work on cold reading and leading the person you're reading by using suggestive speaking; tarot are designed to be vague so that the reader can make blanket statements, but in such a way that the person feels it applies directly to them, even though it could apply to anybody. Practicing cold reading can make you very effective with tarot, as you'll be getting the person to volunteer personal information to you without them realizing it, and then you will essentially repeat it back to them later, which will impress them. It's easier with people who want to believe, but not so much with skeptics. Haha.