The domain name should always be first! The right way is to purchase your own domain name first. Be careful and do several name searches.
A domain name that is short and simple is easier for you and any visitors to remember. Most hosting sites have very good searches for the name(s) you might desire. The domain name does not have to be on the same hosted site as the domain name is purchased. Important: when you own your domain name your site will be known as myname.com or whatever you chose from available names. You may not want myname.atmyhostname.com. The latter is a subdomain at some other hosted site. Like this blog is blog.applicationallstars.com meaning it is on the same hosted account with our site.
The hosting is very important! If you are using a developer, listen to the suggestions he makes for a hosting site. There are sometimes reasons other than price that will cause the developer to suggest a particular host for your website. He is better equiped to deal with the nuances of that host, price could be a factor, type and style of your app, as well as other reasons. I personally build most sites on Namecheap hosting for linux type applications and Smarterasp for the windows type applications. They fulfill what I need for the different type applications. There are hundreds of others that are very good to deal with out there as well. By choosing a top hosted account, you will have all the tools to install other applications onto your site like this blog or out forum. There are many really good free ad-ons you can get on a good host plan. Blogs, forums, Image Galleries, utilities, and many more. If you are going to build your own site, there are free tools to do just that. Komposer is a free tool and does a great job with websites for both the beginner and the advanced user, but there are many more. Bottom line. The cost is always a factor, but a domain name is about $10.00 per year and a Hosted account can vary from a smaller space at about $40.00 per year to untold amounts, but for about $50.00 a year you can have your own spot on the world wide web that belongs to you and you can do what you want to with it.
The build your own free site online like Wix, Yola, Weebly, and others and the Potential Problems: Not to be knocking anybody's services, but you should know a few things.
There are some very real reasons to NOT do this and they may not apply to you. I will attempt to document a few.