Grow your own site: the web host
The proliferation of different web services can make it difficult for someone who isn't focussed in that area to decide which to use. In fact, quite a few of these problems can be solved by deciding
- what you want to display; and
- where you're going to put it.
The where is the domain [heh] of the web host.
A site is just a collection of specialised documents or files. In basic terms, the web host is the entity that holds and displays those documents/files for you. Sites like WordPress.com are in fact web-hosts, but specialised: it has its own system for managing and displaying your content, and various themes. But if that's where your content is, then they are your web hosts.
As an artist, there will be a number of considerations.
Ease of use
How technical are you? How technical do you want to get? It may be that you use a service like wordpress.com or tumblr, and then buy a domain name that points to the site—if the most important thing is for the web address to say "myname.com" then this is a perfectly reasonable solution.
But if you want to—or need to—control all aspects of the site's design, then it's worth knowing if the web host will allow you to install the things you want to use. My personal preference was to build from scratch—but I have a web host that allows me to do this. In the ten years that I've been using the web, the resources to learn web languages have improved manifold; and it's another useful skill.
Whichever way you choose, I must stress the use part of "ease of use". You must be able to modify, add and delete the content yourself—and be willing to do so if required. Having "someone who does it all for you" is filled with pitfalls - what if that someone goes on holiday and you want to put up a national tour? Or change a date of an exhibition? It's just a basic responsibility thing.
It's worth considering how much space—in terms of gigabytes - you will need for the media you wish to upload: but also consider what other services you're putting somewhere else. For example, if you've already put most of your music onto Soundcloud [or ccMixter, or Bandcamp, et al.] then you might be able to get by with less space to host your site.
Some web-hosting services use adverts to balance out the cheapness of the service. I cannot in any way recommend this for a site about anyone's work. Decent web-hosting will cost - not a lot necessarily, but it's an expense that you will have to factor in unless you know someone who will happily give you the space for free.
Luckily this gives you the chance to shop around, and get the best deal for you. So take some time and do it. Look at what they offer you. Will they help you get the domain name you want? Do they charge extra for hosting MP3s [don't laugh]? How reliable are their services? Balance this with what you can afford.
The law is different in different countries: what's all right in the United Kingdom might not [well, probably would not] be fine in the United States, so it's worth using a host in the same country as you. If things go horribly wrong in the relationship from a legal or technical standpoint, it will be easier to sort out. Shop local!
Where you put your site says just as much about you as what you put in, so it's worth taking the time and effort to picking one that suits your needs.