Start your search by thinking about the type of blog you want to have and how much control you want to have over the look and domain name. Free blogging services usually have a set of standard templates and easy controls to get started, but the design is usually not terribly flexible. Paid services often offer more design control, feature management and tools.
If you are an artist or a photographer and want to quickly post about the things you see, a visually-oriented platform like Tumblr may work best; Tumblr is free and has its own mobile apps for blogging on the go. If you want to compose long and thoughtful essays, a blogging platform suited to longer writing and editing, like Medium, may be most useful. (While Medium has a mobile app for iOS, it is currently intended for reading material on the site rather than writing.) Some platforms like Typepad and WordPress are flexible enough to handle either approach.
WordPress comes in two versions. One is the free WordPress.com site, which provides less customization but hosts the blog on its own servers and has its own mobile apps. The other version is found at WordPress.org, where the WordPress blogging software can be downloaded and used on someone else’s servers. You have more control over the look and feel of the blog there.
Some free services, like Google’s Blogger, make it easy to set up a basic blog, especially if you already use some of the company’s other products, like Google Plus Photos. Paid services like Squarespace may cost $8 a month or more, but you get technical support along with perks like search-engine optimization and slick designs that work for both desktop and mobile readers.