As a part-time web designer and full-time amateur musician

... you have 3 viable choices:
1)
The cheap/complex way: if you know your basics in HTML, you can make yourself a decent site. I pay $2.50 a month for my website (25GB of storage). Not too shabby. Although building/updating the site can be a bear.
2)
The more expensive/simple way: use sites like wix.com or bandzoogle.com that make building the site really easy and user friendly, BUT keep in mind these companies up the hosting cost and lower the storage size. The good thing is you'll have a sweet looking site, but you're gonna spend over $100 a year on it.
3)
The free/stock way: use sites like wordpress.com or blogspot.com and host your site for free. These are user-friendly sites, BUT they take away from the "personal domain" look and feel -- with their logo and ads on your site, it makes it look a little less professional, BUT IT'S FREE. And something is better than nothing. I started gaining traffic to my site by using Wordpress while I built my website using HTML. Sites like Wordpress are great to see user-generated traffic and create a "subscription"-powered fan base.
You don't have to pick just one choice. For instance, I use a hybrid of both 1 and 3. With the power of Wordpress I can easily update my fan-base without having to do any coding (I call it my Blog). But having my own domain name makes it easy for people to find my site, since it's just
MYNAME.com. For awhile I just had my domain-name redirect to my Wordpress blog and used Wordpress as my primary website.
Sounds a little confusing, but I think once you outline your budget and gauge your technical skills you'll find some well-off way to host your music.
Good luck!