What are they in now? If styrofoam, what I do is get the new pot ready, with a hole in the middle to accept the plant+soil, spread some rooting powder around the edge of the hole, then use an exacto knife to slice down the side of the styrofoam cup which makes it easy to pull out the whole soil mass intact. Plop in hole, even out the dirt, add a bit of water, and you're set. Note: wait until the plant is dry before doing this, since the dried soil mass is much less apt to crumble.

If it's not a styrofoam cup, then what I do is let the pot get dry, prepare the new pot as above (hole, rooting powder), then use a butter knife to slice down the sides some (but not too far), then flip the pot upside down with one hand, while holding the stem between two fingers with the other hand, and see if the soil mass slides out. If not, I tap the bottom a few times and see if it comes free. If not, turn it back over, reapply butter knife a little deeper and repeat until it comes out. The deeper you have to cut, the greater the chance you'll damage some roots, but I still think it's better than just digging in with your fingers around the stem, because that pretty much always causes damage. If the plant is healthy, it'll recover either way, but I think the butter knife approach and tapping out the entire soil mass does the least overall damage.

I also personally only transplant twice. From styrofoam cup to 1/2 gallon square pot, to 4 gallon square pot. Sometimes I skip the 1/2 gallon pot too, but the plants do better if I don't skip that one. So I have never transplanted from any pot larger than a 1/2 gallon pot, so what you do if you want to transplant from a 3-gallon to a 5-gallon, well, can't help ya there, never done it.