Depending upon the scope of your web site, you may need to choose two different companies. Building a web site is a highly technical process. Designing a web site is a highly creative process. Many advertising firms specialize in web site design which does not necessarily require any web development skills whatsoever.
The process of creating a web site is similar to the process of building a new home. Before you ask a construction company to start building, you first seek out an architect who creates a blueprint of your house taking into account what you want (number of stories, square footage, etc.). Creating a detailed blueprint before construction begins can help you accurately estimate the final price. Without the blueprint, you may end up paying a lot of money for a house that does not fit your needs.
Creating a web site is exactly the same except most web site "builders" also claim to be "designers". The good news is that you can look at other sites a web design company has created (like looking at other homes that a home builder has made). Make sure you ask the web design company what their process is for designing a web site vs. building a web site. They should understand the difference between these two concepts. If they don't, they're probably builder that think they can also architect.
