There are two different types of web design.
The look of a page or site, the colours and shapes and graphic styles are all what most people think of as design.
In addition these is design for useability. This means the design of the navigation, searching and display to make it as easy as possible for the visitor to find the information they want.
There are a number of good designers in almost any area. Unfortunately there are always a few bad ones too.
In order to choose the right designer, we recommend you ask to see some previous examples of their work. Question them closely about which bits they actually did themselves. Also ask about which technologies they use. They should be telling you about search engine compatiblity, ease of navigation and adherence to guidelines on accessibility for people with special needs.
Work out the rate they are charging in terms of hourly rates.
Sometimes the two sorts of design are in conflict with each other and you need to find designers who understand this.
For example 1 in 12 people have some sort of color blindness. When you're designing for the web, this means that 1 in 12 people might not be able to see your site if you choose a colour scheme that is known to cause problems for color-blind people.
For a web based business this effectively means losing alomst 10% of potential sales. Suddenly web design becomes a very serious business.
Here are some links to sites which give good advice on both visual web design and and useability web design.