I’m very new to Blender and I can’t figure out how I could (easily) connect the two shapes in picture 1. I want the outside to look like picture 2, but I don’t want it sticking through on the inside like that. How can I do this?
Hey, blender comes with an awesome modifier that’ll make this very easy for you. You just shrinkwrap the right cylinder’s most left vertecis onto the left cylinder and then apply it (shrinkwrap mode: project, and only y-axis here)
Jonathan Williamson made a nice little tutorial on this on blendercookie.com. Just don’t forget to actually apply the shrinkwrap modifier before deleting any faces as he did the first time 
The most quick and dirty way though, would be the boolean modifier.
If you’re using 2.5 there is even a Pipe Joint addon to make life really easy.
And no, I generally wouldn’t recommend the boolean modifier.