Any good web developer/designer will normally have several browsers installed on their computer(s) for testing. I myself use Internet Explorer (obviously), Firefox, Mozilla, NetScape and sometimes Opera.
Once I have confirmed a site in those browsers I often check on one of my Apple Macs to see how the pages display in Safari and if I hapen to have a bit of time on my hands I sometimes check them in Konqueror on a Linux computer.
In terms of testing your web programs, any of the mainstream browsers should be able to cope with that unless you start getting into some pretty scary coding. Without getting too technical there are certain coding functions that could potentailly cause a browser to trip up if you try to run them on a poorly set up local testing server, but if you are learning web development you are very unlikely to be calling anything odd enough to make things fall down around you.
The short answer is that no, you don't absolutely have to have IE but part of good design for the web is cross-browser testing, so at the end of the day you will have to test in IE at some stage.
My advice is to take the time now and make certain you have a reliable development platform BEFORE you start programming for the web. Once you get everything installed to your satisfaction I then suggest you take a hard drive image using any of the drive imaging programs (Norton Ghost, Drive Image, Acronis True Image etc) so that if anything goes wrong you can recover everything, and I mean everything, within a half hour or so.