Since your final output is for the web, PhotoPaint would be a better option then draw. This will give you more control over the background color/texture options. However, in Draw, once you type the letters (and it remains the selected item), all you need to do is click a color swatch on the left hand side, and it will change the fill color of the letters. If you right-click a different color it will change the outline color of the letters. In order to get a background color, you will need to create a box (or any shape) and color it the background color in the smae manner above. Then you need to move the background box over top of the text, right click the box, and select "Order" from the right-click menu. Select "Send to back of layer" and it will appear behind your logo. Click and drag (with the select tool - the arrow) around both the box and the letters. Go up top to the "Arrange" menu, and select the Verticle and Horizontal center options (or with both items selected type "C" then "E").
Now your logo is done, select "Export" button, and export the image as a *.jpg - be sure to check the "Selected only" check box (make sure everything you want in your final image is selected) and save the logo. It will prompt you for color and resolution. Select 72dpi for resolution, and 8-bit palleted for color.
Hopefully this will get you under way. This will be a very quick and dirty logo, and as mentioned, Photo Paint would be the ideal tool to use. The procedure would be very similar, but in order to change colors, you need to select the text, and then highilight (click and drag) the letters you want to change the color of. Also, when changing the colors of objects (background boxes) you must use the "Fill" tool (looks like a paint bucket). Otherwise, the procedure is very similar. The main difference is under the File Menu, there is a "Optimize for Web Export" option. That works much better then Draws export feature.