Q:

Is the wiring in this subpanel correct?

Just want to make sure the wiring is correct. I have seen examples of the of the black and red switched in the sub-panel. This is a temporary connection to the subpanel using double 60 amp breaker with 6 gauge wire. New 200 amp service will connect the new box and the old box will come out.
Did I wire this correctly?

A:

If you're in the US, this is NOT correct.
Enclosure not bonded
The bonding strap on the right side of the panel (when looking at the panel), should be moved to the left side. It should be connected to the enclosure using the green screw, and the tab should be connected to one of the grounding bar's terminals.

Double tapped neutral bar
On the neutral bar, only a single conductor can terminate per screw terminal. You've got two wires per terminal, which is not allowed.

Neutral and grounding conductors not separated
In a secondary panel, the grounding conductors and "neutral" conductors must be kept separate.
You should move the grounding conductors, and terminate them at the grounding bar instead.

Neutral and grounding bars not separated
You should also remove the bonding conductor that is connecting the grounding and neutral bars together.  (Thanks @ArchonOSX).

Unprotected wiring
I'm not sure what's going on at the top of the enclosure, but it looks like you've got some conduit work to finish up.

NOTES:

The main panel photo is fuzzy, so I can't tell what size breaker is feeding the second panel. You also didn't mention the size of the conductors between the panels. So I can't say for sure if you used the proper size conductors.
It also looks like you connected the ground to the neutral bar in the main panel, and the neutral to the ground bar.  Not such a big deal, since they're bonded in the main panel.
This answer is based on National Electrical Code 2014.

A:

This does not look correct to me, assuming this sub panel is in the same building as the main panel.
First of all, the neutral and ground should be separated, but it does not look like you have removed the bonding strap.
Similarly, the few circuits you have wired do not have their grounds going to the grounding bus.
If you are not comfortable installing a sub panel it might be a good idea to hire an electrician.

