Can I skip the ender dragon and go straight to the outer islands?

Yes, it is possible. Yes, it is very useful to do it (despite of what the other answers seem to suggest). And yes... it will be a painful experience.

It's worth it early in game, when you're still not ready to take down the Dragon, but really want an elytra and ender chests. Plus, the Dragon fight is much easier if you can fly.

How to get there?

The outer islands are in a ring with a radius of 1000 blocks centered around the Dragon's island. So, once you spawn in the End, you'll have to make a bridge of approximately 1000 blocks in a straight line. Remember to crouch so you don't fall!

For the first 200 blocks or so, you'll still be in range for the Dragon, so she could potentially attack you and throw you off into the void, resulting in your death and loss of all your inventory. I have done it twice in different worlds and never had this problem, but maybe I was just lucky.

And then what?

Once you reach the outer islands, you'll want to find and End city (unless you're only interested in chorus fruit? :). This can be very painful, the nearest city can be still very far away (2000 blocks further away in my own experience, no idea if there is a limit). But at least now there is solid(ish) ground below you, and you can mine some end stone to keep building your bridges between islands.

A couple of suggestions for this part:

• First, bring with you a pumpkin helmet to avoid angering the endermen. You can do it without, but you'll be on the edge (quite literally!) every time you bridge between two islands: one bad timed attack from them, and into the void you go. Also, if you're playing on hard difficulty, they hit pretty hard.

• And second, if you get tired of wandering the End and can't find a city, you can use Chunkbase or other online resources to find the coordinates for the nearest city in your seed.

Finally, as the others already pointed out, if you don't kill the Dragon there won't be a portal back to the Overworld, so you'll only be able to go back by dying. However, you can still keep all the loot, which is what makes this really useful for early in game. You just need to put everything (including your armor, elytra, and offhand item) in an ender chest before dying.

So, what to bring with you in this expedition?

• A bed to set your spawn in the stronghold before jumping in the portal.
• An ender chest to place in the stronghold, and another one to bring with you. So you can place all your stuff in one just before dying, and get everything back from the other one afterwards. There are ender chests in some cities, but I wouldn't rely on finding one, better to bring it with you.
• A silk touch diamond pickaxe, otherwise you won't be able to retrieve your chest once you place it down to use it.
• A pumpkin helmet to have the enderman ignore you.
• Some regular chests and a crafting table, so you can build a few shulker boxes right away. Otherwise you'll only be able to bring back up to 27 items, which is the ender chest capacity.
• Rockets! You'll want to use your brand new elytra, right?
• At least 1000 building blocks for your bridge. That's 16 full stacks, but it'd be better to take 20 stacks, just in case. You'll need more than that, but the rest you can mine in the outer islands.
• Wathever you need to fight the shulkers: a bow, a shield, feather falling armor or a bucket of water...

Good luck!