I have a few photos where the person's arm, shoe or something was off my back ground.  I'm still VERY new to photoshop and right now I'm playing with the clone and patch tool but it seems like there has to be an easier way. 

Yes, I know the easiest way is to make sure everything stays on the back ground for the few that wants to give that answer, sorry I beat you to it.

What I'm doing is going over the area with the clone tool and then going back over the area with the patch tool.  Is this really what I need to be doing or am I going about this the slow and wrong way?

Without knowing exactly what you're trying to put in, I'd say that cloning it in is a good start.  Depending on what it is, you might benefit from actually compositing in segments from other photographs that share a relationship with the picture.  Do you have photos from the shoot that have some aspects that you can bring in to make a repair to the photo you're working on?

I'm hoping I'm doing this right, but here is an image that I started working on, so I'm hoping I'm doing it right but some how I can make it look better.

I am not sure how you are doing it . There are a few ways to do it. One is the way you are doing it. The other is to create a duplicate layer, and paint and then add a layer mask and work with it. If I understand correctly you are trying to add the blue backdrop to the whole image?

I see what you have.  Use layers and masks.  The patch tool is a little cumbersome for this.  I would make a copy and on that copy, I'd make background in the areas I need it, ignoring the kids.  Then I'd mask that layer, and there are a lot of ways to do that, and that's a whole subject area by itself.

If you get fed up with it, I'll fix it for a small donation to the Feed Me Fund.

Make a selection of a good section of the background. CTRL+C then CTRL+V. This gives you a new layer that contains only the selection you made. Use your move tool, grab it and move it into place. Add a mask. Paint black on the mask where you don't want the background. Rinse, repeat if necessary.