Finding a 'web designer' is easy. Throw a handful of gravel and you will hit at least 10.
Finding a _good_ webdesigner who knows UI rules and how to design a site that fits your budget, your functionality needs, and who knows good design in general isn't so easy.
Some tips...
Shop locally, and go for smaller shops who will treat you like a person instead of passing you off to their flunkies. THe smaller guys treat you right, and are almost always priced better than any agency.
Never look at a webdesigner's site as an indication of anything. Most of them put tons of effort into their site, thowing in all the bells and whistles, and have an mostly endless budget to create something that is a display of chest-beating and loud growling to let you know they have skills. This is a common mistake by (usually amateur) web designers who think that over the top fluff will get them work. I can tell you from my many years of experience that it will run off more business than it gets you. Clients like Nike and MTV who have millions to spend do not exist for the real world people like me. Instead my clients are budget minded, smaller companies who depend on their site to load quickly, be powerful and easy to use, and not overwhelm their customers with flashy fluff. Especially when a customer is there to find and purchase a product.
Instead... Look at their portfolio of real sites, follow the links to those real sites. Ask them questions about the budget vs quality of work, why a project was what it was and how it came to that point. Every site has a story that explains why it looks and works like it does. Sometimes the client has no money and you have to skimp, sometimes the client is uber picky, sometimes the client hands you a sketch and expect you to reproduce their horrible design. And these things reflect on the designer, whether we like the end result or not.
Pick a designer who knows to check your site on ALL browsers, from AOL, Netscape, Safari, Firefox, IE, multiplatform Mac and PC versions of these browsers. Any designer who depends mostly on IE is a designer to avoid. If he poooos your questions about his ability to get the same look on IE, FireFox, Safari, etc, walk away and don't go back. Remember that your customers will not all be using IE 6 and many will not come back if they feel their browser choice is not relevant to you. I'm sure you have seen these sites before, they even go as far as putting "This site best experienced using Internet Explorer" on the home page.
This next one I feel very strongly about and I personally always make sure to talk to the client about it. Web designers who put their name or company on your site. It is common for some designers who think they are all powerful to put their names and link to their business on your site. Many will not even ask if they can, they just do it and build that idea into their design of your site. I only do this if we discuss it upfront and I discount their site as a result of my link. If it's charity or a buddies business and you share and share alike, fine. But otherwise this is a no-no.
Finding a web designer... Aside from everyone in this thread who has kindly donated their name to your hat, I still recommend local. Face-to-face is very important to me and my clients, but if you don't mind never seeing the designer that is fine. But when you go local, you can goto your local Rotary or other groups, ask around, check with other local businesses to see who they use and/or recommend. Check the phone book, cross reference names and look up their work.
Why should you listen to me? Well, I have been designing in print for over 15 years. And I have been doing web design for almost as long as the web has existed. I have watched fads in design and functionality come and go, and always tried to steer my clients away from fluff that would make their website junk.
I hope I have been helpful. And I wish you luck in finding the best designer for the job.