What is the easiest way to get up to a floating island?

The easiest and safest way in your current state (I think you are not in the hardmode) is to use a Gravitation Potion, which can be easily found in pots or chests at the underground level. You may have got one by now with all the mining you did or you can craft it with some ingredients, but I don't think you got a feather at this stage into the game

The easiest way in early game to climb up and down fast is using ropes. You can buy them from the merchant for 10 copper each, or find them by smashing pots in the caves. It goes without saying that this item is very common and very useful until you gain access to movement-based accessories (boots, wings, etc...)

You can also place blocks around the ropes, so the eaters of souls can't cross them, and you'll be safe going up and down with the rope.

But the safest, (and most expensive) way to get up is to use teleporters. They are only accessible in hardmode though, but once placed and wired up, they will transport you in your island in a blink of an eye.

I guess easiest is subjective -- because to me, the easiest would be using only materials I have already and doesn't require going and getting anything else -- ie "nerdpole" up to the top. Just jump and put a block under you, and repeat that until you get to the floating island. Granted the way down becomes much more complicated, but it does allow getting up relatively quickly.

Aside from that, I've used a slightly different technique that actually creates jumpable "steps" with the dirt blocks that I can jump up to, then later run down without taking injury. It may be slower than using ropes, but if you don't have ropes already (or can't make them) you can get plenty of dirt blocks.