I can only go to Object Mode I can't go to Edit Mode or any other mode besides Object Mode. 

Can someone help me out? It won't show any other mode besides Object Mode.
There are mainly two ways to switch between Object and Edit mode.

1. TAB


As @depperm suggests you need to have at least one object selected;
you also need to have the mouse pointer inside the 3D View window;
just press TAB to switch.


2. Mode menu


In the 3D View header you'll find a mode menu (see picture below);

just click on it and select the mode you want to switch to.