When developing for the web, it's important to check your website using several different web browsers. Microsoft's Internet Explorer (IE) is notorious for displaying or interpreting your web pages in ways that are slightly different than most other web browsers.
Firefox Very common, and so far less prone to hacking than IE
FireFTP Use this Firefox add-on or the standalone Filezilla (below) to transfer
files to your website
Google Chrome This has become my second browser of choice. It's more streamlined than IE and Firefox
Safari (Mac and PC) Apple's browser for the Mac, but there is a Windows version that's pretty cool
Opera This Norwegian web browser has been popular in some circles for quite a long time
Notepad++ This open-source programmer's text editor color codes your HTML and CSS (and other computer languages) and has a lot of nice editing features.
Filezilla FTP client software for transfering files to your website.
Yes, you can spend hundreds of dollars for Adobe Photoshop, but the following two free image editors do about 90% of what you ever need for basic image processing. They're fast, too. Although I own Photoshop, I prefer to use Photofiltre or Paint.NET whenever possible. I only pull out Photoshop when I absolutely need one of its features.
Photofiltre nice, fast image editor for most common tasks
Paint.NET more features, and allows multiple layers
GetThisSoftware.net links to more software recommendations