Yes, you should have a website for your illustration work. A website functions as an online portfolio and is a vital necessity for illustrators today. Whether you design and maintain the site yourself or hire a professional you need to keep a few things in mind.
A professional website is for your illustration work only. Any private, personal or non-illustration information should be kept on your personal website. Make sure your professional site is well organized, well designed, quick to load, and easy to navigate. Avoid flashy graphics that slow down and distract your viewers from your work. Keep the site updated with new work while taking down old or outdated pieces. Buy your own domain name instead of relying on free sites that display advertising.
A blog isn't necessary but if you have the time and interest in maintaining one it can be an interesting addition to your web presence. Here are some examples of illustrator blogs: The Dust of Everyday Life Drawn: The Illustration and Cartooning Blog DrawerGeek (not a traditional blog but more like a blog of drawings!