My understanding is that it depends on whether or not it’s a “new world” or “old world” tarantula. You generally don’t handle old world tarantulas because they can be feisty and tend to be faster and more venomous than new world tarantulas.

New world tarantulas are generally pretty chill. As long as you’re gentle and patient, they’ll typically tolerate you picking them up. The way I always did it is that I’d slide my hand underneath mine (which can be a little scary the first time you do it because they may threaten a threat display, but as long as you’re slow and gentle you’ll be fine).

If you try to move your hand and they’ve grabbed onto it and another object, stop! Don’t panic! You’re fine, just bring your hand back to where it was. They may be still trying to decide if they want to ride on your hand, or if you were just holding them, trying to decide if they want to get off. If they seem to be stuck, like on clothing, slow down, stabilize them, then gently stretch and relax the fabric around their foot until they move it.

Make sure they’re never more than about 6in/15cm above a surface. They can survive higher falls (the softer the surface, the further they can safely fall), but they can and will fall off, and they can go splat if they fall too far. 6in/15cm gives you some breathing room. Don’t try to catch them if they do fall (unless it’s an obviously fatal height), let them catch themselves. If you do try to catch one and you don’t kill it in the process, you may find yourself with a very scared and/or pissed off tarantula in your hand; so be prepared to get bitten and have your hand itch for the next day or two (though imo it’d be worth it if it was about to fall to its death as they can live a very long time in captivity).

On the subject of itchiness, I’d recommend gloves the first couple times you handle one. Both old world and new world tarantulas have urticating hairs which can brush off onto you and make you itch as a defense mechanism. Additionally, new world tarantulas can actually flick them, so try to avoid getting your face too close.

Don’t let them crawl on your body. Not because they’ll do anything bad (except maybe poop), but because they’ll happily explore this amazing new landscape and may decide to try and hide in your shirt, climb onto your head, walk around on your back, etc. Basically, they’ll go anywhere their little feets can take them, including places that are hard for you to reach (like your back) or can cause a panic response (like your head).

Don’t let this discourage you if you want to handle a tarantula though! Treat them like little 8-legged babies with brittle-bone disease and you should be fine.

Edit: and as always, don’t try to handle a tarantula that doesn’t belong to you. Their owner may have specific rules for handling them, so make sure you ask about them if they offer to let you hold them.