Design is like any branch of the Commercial arts - it's all about problem solving.
First, you have to define the nature of the problem:
Mast heads (logo is actually short for logotype which is something different to what people think) and site designs are all about telling the world who and what you are. They are statements about how you see yourself and how you want people to perceive you.
The typeface, the colours, the style, the layout - all affect the viewer's perception of you.
This is why big companies spend a fortune on brand image - and why every aspect of that image is precisely defined.
So now you need to define the problem itself:
What is wrong with your old site and mast head?
And it is not what you think that matters but what your target audience thinks.
Then you need to define how you want your audience to perceive you.
This will determine colours and style.
Then you go on from there.
In Design asking the right questions is important because the answers shape your solution.
And remember the golden rule: keep it simple!