In Word 2007, when you first open the text box you get a new tool bar which allows you to change all the formatting of the box and it outlines. However, I cannot find a way to get back to this formatting toolbar once you start using the text box.
Knowing the version will help. There are so many versions being used, I think, and they all seem to be so different. HB has 2000, xania has 2007, I have 2010
Having said that, HB's post covers all these version, ie Right click on the text box to get a menu, in 2010 I click on 'Format Shape', then in 'Fill', check 'Solid Fill' is selected, then in the box 'Fill Colour' click on the Colour box and choose your colour. On my PC, I have the option re Transparency as well. Then click 'Close'