Portfolio
The single most important thing to look for when selecting a web designer is to look at their portfolio. This will tell you all that you need to know in terms of the type and style of websites they have produced in the past. If you like the look of their past work, then you're ready to take the next step.
A good portfolio, as well as being visually striking, will also boast a good variety of different projects, and should almost certainly feature some which have been completed recently.
Brief
Before you approach anyone, it is important you have a clear idea in your own mind about what you want your site to consist of. Do you need to sell anything online? How many pages do you need? Do you have an idea of how the site should look? Do you have a deadline? Try and get as clear an idea as possible about the whole site.
When it comes to discussing the project with a web designer, they should be able to help develop your idea either by making further suggestions about what might work and what won't. By the end of it all, they should be in a position to give you a price for completing the work, although this may not be instant. In all likeliness, they will take away your brief and then come back with a detailed quotation.