If you click on your desired Object, you should see a bounding box around it, as seen in this picture.
When you have your desired Object selected, go to the second Editing option on the window to your right that says "Modify" and click on it. I put a red box around the Option you should choose.
Then you should see your list of Modifiers under it. It should say either "Editable Mesh" or "Editable Poly".
After that, click on the + icon next to your modifier, and select the Vertex Option.
You should now see the vertices that your selected object has. The vertices should show up as blue squares/dots on your model.
To edit a vert, just click on it. But this alone just selects it. A selected vert will show up red.
And if you want to select more than one vert, then hold down your Ctrl key, and click on all the vertices you like. And if you accidentally select the wrong vert, just hold your Alt key (you're safe to let go of Ctrl) and click on the vert that you didn't mean to select.
Right-click, and you're able to choose between "Move", "Rotate", "Scale", and "Select". Each option should be self-explanatory.
Make sure you make a Back-up of your file and make sure that no other Modifiers are in your window. If there are, just Right-Click in that area, and click on "Collapse All", and accept the warning prompt.
This is just the basic stuff, though. Hope this helps.