Hi, I'm using Maya 6.5.
You're new to Maya ? I think I can help.
Well, so far I've tried the standard texture only. If you want to export textures to .egg, I think it's the only way, using the standard texture.
First, you can open the Hypershade window, and select the material you want to texturize, and then the attribute editor will display the material attributes. Look at the 'Color' attribute, if the material is just a color, then at it's right side is the checker box (button with black/white pattern, like the chessboard). Otherwise, if that material has already texturized, then the button will be '>' symbol.
If you want to apply texture to that material, then click that button, and the texture panel will pop up. Make sure to set the texture option as standard texture, not projection, stencil, or other option. The radiobuttons are located at the top of the texture panel window. Below those radiobuttons are the texture assignment buttons. Click the 'File' button, it means that you want to use external texture files.
After doing so, new attributes are added to that material, which are the file and the texture placement settings attributes. You can see two tabs of those attributes in the attribute editor. Bring up the File tab, and click at the browse button at the right of the filename box to load an external texture file.
Then you can set the texture placement settings. Click the other tab in the attribute editor, and you can adjust those settings as you wish.
Before you apply a texture to polygon, make sure you have set it's projection methods, or the texture won't be applied. The projection methods are planar, spherical, cylindrical, or auto projection. Before you assign a texture, first select the object you want to texturize, and select menu Edit Polygon>Texture Projection> and select the appropriate projection method suitable for your object.
PS: read the manual of Maya about texturizing, coz it is the most basic thing to set the appearance of your objects.