Currently there are opportunities to register a free domain name that ends with co.cc. You might consider it very similar to registering a subdomain, but with the added value that it is similar to co.uk, co.nz or co.in. The difference with co.uk, for example, is that you'll be charged around $9.99. You might actually find a domain name that you like for your subdomain at freedns.afraid.org.
After registering your free domain, look into free domain hosting, such as with 000webhost.com.
One thing that is very important to consider, however, is what you want the website for. I would say to go through the free route so that you can gain experience in managing your own domain. If you want to monetize your website, then you should definitely register a domain for a fee.