First you need to decide what type of website you want. Is it going to be a showcase for your work, a contact page, a store, email, or everything and anything.
If you're making a store page you're going to want a host that is going to provide you with good: storage, bandwidth, security, databases compatibility and stability.
Storage:
Unlimited. Lets you use as many pictures, text, products, and video that you can dream up
Limited. Pay $ for x amount of space
Bandwidth:
Unlimited. Let's 1 person or 7 billion people visit your site as often as they like
Limited. Pay $ for x amount of data transfer. After you use up your available monthly bandwidth your site is shut off until the new month, or you're charged for the amount that you go over.
Security:
SSL = Secure Sockets Layer. You need SSL if you want to sell stuff on the web and keep customer information safe. Most of these will require that you get a Unique IP address for your web server.
Database:
Databases are essential for e-commerce sites, unless you sell via paypal or the like
Stable:
Picking a stable host insures that your website works when someone goes to it;)
I use www.dreamhost.com for all of my sites and for my clients. If you have a specific question about hosting or dealing with a webdesign person/firm feel free to PM me. Take your time and do some research. Hosting prices barely change year to year. If you see any kind of sale or "Buy before it's too late" type thing, chances are it's just a sales tactic and not a deal.