What is the best DDR3 Memory Upgrade I can to? PC3-12800 or PC3-10600

Ram is sold in kits for a reason.
Ram from the same vendor and part number can be made up of differing manufacturing components over time.
Some motherboards can be very sensitive to this.
That is why ram vendors will not support ram that is not bought in one kit.
Although, I think the problem has lessened with the newer Intel chipsets. Still,
it is safer to get what you need in one kit.

Your ram will operate, if at all, at the lowest common denominator of specs.

Considering that ram is cheap, I would replace them all with a single kit made up of two sticks. That will be 8gb or 16gb.
Two sticks are easier for a motherboard to manage.

You want documented ram compatibility. If you should ever have a problem, you want supported ram.
Otherwise, you risk a finger pointing battle between the ram and motherboard support sites, claiming "not my problem".
One place to check is your motherboards web site.
Look for the ram QVL list. It lists all of the ram kits that have been tested with that particular motherboard.
Sometimes the QVL list is not updated after the motherboard is released.
For more current info, go to a ram vendor's web site and access their ram selection configurator.
Enter your motherboard, and you will get a list of compatible ram kits.
While today's motherboards are more tolerant of different ram, it makes sense to buy ram that is known to work and is supported.

I would buy a 1333 MHz dual chanel kit but I see that it's been already mentioned and very detailed.

Okay so on the compatibility list most of the ones are 10600 or 10666 at 1333 mhz. I was told that higher the number, the better. As in 12800 or 14900 are better/faster in performance. Is this not true? How come my computer has each kind, but motherboard compatibility list is limited to only one or two kinds of RAM? Also if I were to buy the recommended RAM, but lower down the Latency from currently 9 to 7, is there any significant consequences with regards to my motherboard? What about if I put a 7 CAS Latency kit with a currently 9 latency kit?

Faster ram and lower latencies make the cpu a bit more efficient.
We are talking about perhaps a 5% difference between the slowest and fastest ram.
In general, you will benefit more from added ram than with faster ram.

The motherboard controls how the ram operates. So all the ram must respond to the same voltage.
All the ram sticks will operate at the same speed and latency.