If everything is part of the same mesh, you can go to edit mode by selecting the object and hitting the tab key and separate the parts there.  If all the pieces you want to ungroup are completely separate (no shared vertices between them) you can just hit the p key and select "separate all loose parts."  The parts that are not separate you will have to select individually, hit the p key and separate the selected parts.

I would recommend naming items, it makes them easier to find later, but it is not necessary.

BTW, you can also get out of object mode by using the mode menu on the 3d view window.  In your screenshot there is a white box on the menu bar under your 3d view that says "Object Mode"--that's a drop down menu that lets you select other modes like edit mode, pose mode, uv face select, etc.