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How can I join these two meshes in a proper way? I know I can Ctrl J to join them, but how can I do so without having overlapping faces?

two cylinders in solid shading mode two cylinders in wireframe

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First off, ask yourself if you really need those two pipes connected? In a real hand truck those are two different pipes welded together at the joint. Joining them would actually make creating the materials and shading harder. You could just add a weld around the joint, and you would never even know that they are separate objects (plus it would be more physically accurate).

However since you asked:
Start off by making sure the two pipes have the same vertex count.

  1. Snap the end of the horizontal pipe to the center of the vertical pipe. first step: moving the pipes in to place
  2. Select the horizontal pipe, then Shift select the vertical pipe. Make sure you are looking down the horizontal pipe and then knife Project to make the cut.
    knife project
  3. Delete the new faces, X > Faces.
  4. Join the two pipes in to one object. Then switch your view so that you are looking at the side of the horizontal pipe.
  5. Snap each of the vertices from the end of the horizontal pipe to the location of the vertices in the cut made earlier.
    snapping the last vertices
  6. Select all the vertices and then remove doubles, W > Remove Doubles.
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There are several ways to go about this, you can eyeball it and do it with some clean cuts or use a boolean modifier etc but why reinvent the wheel for rather simple objects when you could be finishing another part of your scene?

Go to User Prefs and enable the Extra Objects addon and you can use the T-Joint object. You can tweak the radius, divisions etc. There are also similar custom primitive types for pipe joints if you're doing some architectural stuff.

enter image description here

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