Starting wordpress blog is very cheap. It can cost as low as $40.00 per year for web hosting and then around $10.00 per year for a domain name.
So you're looking at $50.00 per year to get a wordpress blog up and running.
In the future if and when your blog increases in website traffic then you might need a VPS or dedicated web server to handle the traffic but in the beginning you'll be fine hosting your wordpress blog on a shared web host.
Once you get enough website traffic to need a dedicated or VPS server then you can likely pay for the hosting costs and more with Advertising on your website through google adsense or media.net or even affiliate products and services ads.