The first thing you need to build a website is a domain name, something like dogparty.com. I use GoDaddy.com to buy domain names. They are one of many Registrars, for example, Network Solutions, Bluehost, Namecheap, DreamHost, Hover, plus many more, and any will work. Usually, a domain name costs about $20 per year. Some registrars will include the domain name with hosting. All will try to sell you more products. But at this time we'll just stick with GoDaddy.
Picking the perfect domain name seems easy but most names are taken. Check to see if your preferred domain name is available by going to godaddy.com and typing in the name you want. Make sure you spell correctly. If it's available then buy it. Often you get a discount if you pay for a couple of years. So buy a couple of years. As you check out, be sure to use the correct information for the Registrant Information, basically the owner of the domain name, as it's a pain to change after the fact. Also, you may want to pay a bit more money for Domain Privacy which hides the ownership email and info from people doing Whois Searches.
Make sure you have access to the email you use to complete the domain name registration. Most importantly, remember your (GoDaddy) account number, password, and pin in order to access and manage domain name after you register.
Oh, and when it comes to payment. I always select Auto Renew to avoid losing a domain name. It's expensive to get an expired domain name back. So mark your calendar, keep an eye out for renewal emails which often get into your spam folder, and be sure your payment method stays up to date.