I use AutoCAD as well. There is no snapping to center, end, etc like there is in AutoCAD but Maya does have the ability to make precise moves.

Mainly what everyone has already mentioned but sometimes you will also probably want to move your manipulator on your current object. You can do that by pressing the "Insert" button on your keyboard. That will allow you to relocate your manipulator.

For example, if you wanted to snap the corner of Cube A to corner of Cube B.

Step 1: Select Cube A, press insert, then snap the manipulator to the corner of the cube A. Press Insert again.

Step 2: Select Cube A and snap it to the vertex on the corner of Cube B.

There are also other ways to snap objects. You can create "Locators" and use those as snapping points. You can also create extra vertices on an object to snap to (but after snapping will probably want to delete the extra vertices).

Anyway, the Maya Help Files should have some more useful info about snapping.