Here are the steps and screens I went through creating (another) new Flickr account.
First, go to flickr.com which should get you a screen similar to the following (they change the ad background frequently):
I then clicked the "Sign Up" or the "Join Flickr" button, they both do the same thing, I believe. The next screen should look similar to this:
You’ll need to fill in your name, email address, password (I always recommend a unique, non-trivial password), birthday and optionally gender. I don’t know of anything that uses gender on the site, perhaps it is profile thing. You can, and probably should use an existing email address that isn’t currently associated with a Flickr account.
After filling out the form, click the "Continue" button. When I went through the process making this email, my browser hung, it just kept working. I closed the window, went to Flickr again and it started right up again, asking me to prove I wasn’t a robot. I presume that was a one off error and you should see a screen like this right away:
I was given a simple task, clicking the correct boxes on the screen shown below. I suspect they will offer one of many tests designs to defeat a robot and coincidentally confuse people. You should be able to puzzle out what they want and do the appropriate thing.
Hit the “Send me an Account Key” button and then head to your email. The screen will update, giving you a field to enter the new account key.
Don’t click anything on this screen until visiting your email to getting your account key.
You should receive an email shortly after you hit the “Send me an Account Key” button. Mine appeared as follows:
After getting the five digit code, go back to Flickr to enter the code in that verify screen.
Then hit “Verify” and you should see a success screen:
Go ahead and click "Let's get started" You should receive a Welcome screen similar to this one:
It will be pre-filled. You can change your Flickr screen name or accept their suggestion. When you are ready, hit the “Next” button and continue. If the Flickr screen name is already in use, a common problem, you need to change it at least a bit and try it again. At this point, you’re done and can see any public Flickr pages or do anything else you'd like with the Flickr program.
If you want to give the new account a spin looking at some of my various albums, you can follow this link: J&S Photo Flickr Albums