you'll find that site designers will also be competent graphic designers too, and they'll offer a corporate identity design service. Not all site designers will do this however, so you'll need to ask individual web designers when you first approach them.
Site design has become a good deal more sophisticated than it was years ago, when I first started designing sites for clients. In those days we were expected to do the lot, from concept to completion. Nowadays you'll often find that a designer will come up with the overall concept - layout,colour,style, etc. Then a developer might become involved and handle the functionality,scripts, databases, and so on. I tend to stick to designing now, and use freelancers for the server-side technicalities - it's faster.
Shop around, and get different quotes. You'll find that prices will vary considerably, so always ask to see previous work.