How to attach and join vertices/meshes together?

Hello there people. I am new to this software and may have the greenhorn questions coming up, although I am certain I will be very confident to do things on my own in the future. For now, until I design my beautiful 4 winged fighter jet I always dreamed of capable of brutally marauding the F22 Raptor in both physics and aesthetics. But before This pathway to extreme professionalism begins, I must ask this simple question:

How to attach and join vertices/meshes together?

I know some people have asked this, but my question is very specific with simplicity being the issue:
I have downloaded a barrel that has a tad bit smaller width and height than what I originally wanted, so I bought Rhino (an awesome software by the way) to do me miracles.

I also wish to print it out using a 3D printer so I will be using STL file format.

Here is the picture of the Barrel:

As you can see it’s a simple barrel that I wanted to print out using the 3D printer, I wanted to increase the you see in yellow color by 1 mm in width and height.

I had exploded the entire object and rejoined some of the meshes together to my liking, and then via the transform tab I used the EditBox and increased the inner barrel to 1 mm in both X and Y axis.

After successfully increasing the size, I rejoined all the meshes together,
Now here comes the tricky part:

Although I rejoined all the meshes together I came across something awfully weird:

To print the object, I use CURA from Ultimaker to do me the post-printing procedures. The software is very easy to use, and easily shows “voids” or empty spaces wherein if the printer prints out the object there would be a nasty gap or unfilled space.

Here is the Barrel in Cura without X rays on.

The x Ray shows the areas in which the barrel’s printing process is in jeopardy in red color : nasty gaps/empty space.

And then comes the tricky bit - again.

Weirdly, the area underneath the barrel is not ‘‘empty’’ when rendered, I think it’s vertices on top another or colliding with each other without joining. And that’s why I’m asking how they must be joined.

Now the barrel from the inside facing the top; As you can see, the top area of the barrel that was circled red, is light yellow when rendered. This is because there is a little '‘void’ due to my incremental increase in 1 mm in width and leangth, weirdly I thought I joined them.

If you would understand, I had already joined the meshes together using the join tool in the tool-bar on the side.
But when the CURA software shows red, it means bad news. Please , how can I ‘‘join’’ join them together?

Any one? :confused:

Try running the MeshRepair command in Rhino 5 on this mesh. It will walk you through a series of mesh repair commands such as hole filling.

This might also be helpful for your workflow…

Thanks brian. God bless you man that would be very helpful.