As you know everyone uses photoshop it became a trend but most of us we use it for our websites.
I am going to start by recommending photoshop if you ever need to start a new design for a website because you can use it for initial mockups to show your client a preview of how the website would look like.
I strongly recommend it because it is going to be easier to changes certain bits from a page just by editing the psd file until the client is happy, then you can start coding the website itself.
Another important point would be "Saving Images In Photoshop".
I have always used the "save for web" function in photoshop as this option brings the file size down dramatically and speed up the performance of your website. Also there might be some people that use GIMP because is free. I personally used it for a couple of days but never got the hang of it. So if you are a web developer/designer try photoshop and you will have some amazing results.
I personally use Illustrator just for doing logo's and sometimes banners, flyers and e-shots etc.
The main reason of using Illustrator is because you can create your design in a vector format.
What is a vector format?
A vector format is a never going to lose quality if you scale it down or scale it up. No matter if you want to put a logo on a pen or on a 20 feet banner that logo will not become pixelated.