Domain name is the first thing you should invest in. Those which are free blogging platforms provide you blogs with subdomains, which are not relevant to traffic gain from today's search engine. So if you want to create your online identity from your blog, then you should first buy a top level domain name. You can buy a domain name as much as you think, at a very low price. It can be obtained at a very reasonable price.
If you would like to buy a domain name by going directly to the domain name registrar such as Godaddy, Namecheap etc., then you might not find this cheap. You should take advantage of their discount coupons and offers, which keep coming from time to time. If you buy a domain name, you can also use it with many free blogging platforms such as Blogger.