I begin a project by doing a comparative analysis of other sites that have already been developed for the same subject matter. This allows me to get an idea of what others are doing and to determine what does and does not work. Seeing what the competition is doing can also help in coming up with ideas for something new and creative. The most advantagous thing about doing a comparative analysis though is that you will have a selection of websites that you can show the client to help them (and the designer) in choosing ideas and features they would like to see in their own design.
After getting a few ideas of what the client wants I begin the design process by writing down keywords that I want the users to associate with the design. Such as; clean, professional, grungy, web 2.0, fun, creative, bright, energetic, friendly... This step usually helps rule out a lot of choices so I can focus on creating the best design to fit the clients needs. For example, if the client asked for a clean, bright and glossy website, I can probably rule out grungy textures and dark colours straight away. By asking the right questions and planning early on, you can save yourself a lot of time and frustration.
The next important step is to get a pencil and paper and sketch out some ideas for the layout. Personaly I find it much easier to come up with creative ideas on paper, because when I design for the web on a computer I find myself limiting the design to something that will be easy to replicate with CSS.
I start with small, 30 x 30 millimetre, rough thumbnails and when I have a few I like I draw them a little bigger with a bit more detail. At this point it is good to have an idea of the sort of content that will be in the site to help make decisions for the best place to put the navigation and how to structure the content. Once I have a design I'm happy with I will sketch out a number of variations for the layout of the content itself. This process will help me decide what sort of grid to use.
I have only recently begun thinking about grids, but I think they are important as they can help make the content appear more balanced and they also provide a guide to create a consistant flow of information down the page.
When I have a good idea of where I'm heading with the design I'll open up Photoshop and start developing a mockup. I tend to start working in greyscale, so I can focus on the visual hierarchy, using contrast, font-size/varient/weight and negative space to determine the importance of each section on the page.