This part is really quite simple. Click here to create your account. (Yes thats a referral link... I'd love to get some more credits and don't feel bad about using it)
Once you have registered and get to the billing stage you should see the option to enter a credit card, use paypal or enter a promo code.
At the time of this writing the following codes (in bold) are all supposed to be working:
From what I experienced, you don't have to pay anything to try it out if you have a valid promo code. I'd suggest prepaying $10.00 - $25.00 via paypal though. The $10 credit will get you ~2 months free if you only use one of the smallest VMs. Theres a lot more you can do with these (as we'll get into later) so you'll probably want the extra budget in there if you plan to experiment a lot.
Once you've verified your account and set up some sort of billing or free credit you should find yourself looking at a page like this:
If you don't plan to use SSH keys for accessing your server you can skip ahead to the "Create your first droplet" section. I'd suggest setting up your SSH keys now though to make life easier down the road.