Two Axis Move/Scale/Rotate by Control Clicking
You can constrain the move, scale and rotate tools to only affect two axes, by holding the Ctrl (Mac) key and clicking and dragging the axis you don't want to move. So if you wanted to move an object along the XZ plane, and not have the object move up or down in the Y direction, you would hold Ctrl and click on the Y axis handle to move the object.
Hierarchy Navigation using the Arrow Keys
If you need to select a group of objects, you can click on one object in Object mode and then press the up arrow key to move one step higher in the Outliner hierarchy. For example if you had 'finger' meshes that were parented to a 'hand' group and want to select the 'hand' group, you could select one of the 'finger' meshes and then press the up arrow to select all the objects. In a similar way, if you want to select objects that are adjacent to the currently selected object, you can use the left and right arrow keys.
Save as Maya Ascii to Batch Rename/Edit/Recover Data
Unless disk space is a concern, it is more flexible to save in the Maya Ascii format. The Maya Binary format is a compressed and more efficient file, but you have more control over the Maya Ascii format. If a file is saved with the .ma extension, you can open it in any text editor and view (almost) human readable contents for all of the objects in your scene. In the past I have used this to change Maya version numbers, perform a batch find and replace of object names, and recover objects and data from corrupted Maya scenes.
Double Click in Edge Mode to Select Loop
While in Edge mode, you can double click any edge to automatically perform the loop selection on that edge.
Click → Shift Double Click in Faces mode to Select Loop
While in Face mode, if you have one face selected, you can hold shift and double click on an adjacent face in the direction you want to select a loop, and Maya will select the entire loop of faces.
For anything other than simple test renders within Maya, I would recommend rendering via the command line using the batch renderer. I outlined the steps to do this in this blog post.
F Hotkey for Outliner
You may know that pressing the 'f' key while in the viewport will frame the selected object, but this key also works in the outliner to scroll to the selected object in the hierarchy.