The first thing you have to do in designing your website is to choose one programming language (as php, C#, JSP, Java) and a database (MySql, Oracle or others). if you really want to do it youself or to hire an expert web designer to do this job for you, you can use and customize a free Content Managment System (CMS). In the first situation you can get a free CMS to use for (as Joomla - there are some reasone why you should use it, Drupal or Wordpress), you configure it and you let people know your website.
There are some web development tools (as Dreamweaver, Fireworks, Panic Coda and many others) that can help you designing web pages from scratch and adding CSS, Javascript or Flash jQuery or Bootstrap (any many other useful things) to them. I'll come back to them in another article - for the moment it's important to know they are there and they are very useful. For more information you can always search using Google, Yahoo or Bing. You can use one or more such tools, as you consider best for your project(s). Using Photoshop or software from Corel you can manipulate images to fit exactly to what you need for your website. Firebug is a very usefull tool for debuging a website. It was created as a plugin for Mozilla Firefox, but you can find the lite version for Chrome.
Although I list this step separately, the website should be done throughout your web design cycle (as any other important project). It means that you should develop your website on layers: from the general idea to functionalities and, finally, to the details.
The final stage in designing your website is to test it. It's important that the website to work for all web browserr (or at least the most important as Firefox, Chrome, Opera, Safari and Internet Explorer).
If you want to improve the chances that your website will work in future versions of all web browsers, consider validating the code for all the web pages in your site. This means that you should check that the underlying code of your web page (HTML tags and CSS) have no syntax errors. There are a lot of free web page validators on the internet, use them.
The most important thing, my recomendation: always have an experienced web designer somewhere close. You can always ask for help when you are stuck and you don't know what to do.
