To create a Don't Starve Together Account, follow the steps below:
Ensure that you are able to connect online via your Steam Client. From there, click on the "Play" Button to start the account registration process. A text box will appear indicating that it's your first time playing the game. Click on "Create Account" to begin the process.
Account Registration will temporarily open your Steam web browser (or depending on your settings, your default web browser) - if nothing happens after you click "Create Account", try enabling or disabling your Steam Overlay in your Steam Settings. 
Once your Steam web browser opens, you'll be in the "Account Registration" screen. After you fill-in the required information (including a VALID Email address) and check the appropriate checkboxes, click on the orange "Register" button to finish the process.
*WARNING* When ready to register your account, click "Register" button ONLY ONCE. Clicking more than once can result in error messages.
You should receive an account verification email in your mailbox (or spam, depending on your mail settings) which includes a link to verify your Klei Account. These emails are usually sent out instantly; if you have not received the email after an hour, check your spam/junk-mail folders. Click on that to finish the verification process and go back into the game.
Once you're in the game's main menu, click on the "Play!" button at the top and you should get into the Don't Starve Together Server screen. Here you should be able to join various game servers or host your own.
Don't Starve Together and Klei Account Creation FAQ
Q: Why did I receive an "Invalid Verification" page after clicking on the account verification link sent to me?
A: This could happen due to a couple of reasons, if you waited a couple of days before clicking on the link, it could be that the page had expired. If so, please try the account registration process once more.
Q: I haven't received a Klei account verification email from you; what should I do?
A: Depending on your email settings and/or email service, mails from our servers may have been marked as spam, put into the spam box, or in some cases, deleted entirely. Check your spam inbox, then try restarting the account registration process to get the verification email re-sent. If you are still not able to find it, contact our support team with your Steam ID (number) as well as the email address you used to register.
Additionally, some email services are notorious for overzealous email blocking. If you have another email address from another provider, you can try with that. We have got a rash of these complaints over the past little while, so it seems like something was done with some email services recently that has caused this issue for people.
Q: Steam Servers notification is saying that Steam is Offline; what should I do?
A: Check http://www.issteamdown.com/ to see if Steam is truly down, if it is, you may have to wait a bit until Steam is able to get their servers back online. Steam Downtime is usually under 30 minutes. If Steam servers are up, your local Steam may have logged out or have been put into Offline mode. You MUST be in Online mode to register your account.