It sounds like the switch is at the end of the electrical run; the white wire that's connected to the black wires is carrying current to the switch, then the black wire coming back from the switch is connected to the light:
You need to connect the black wire for the new outlet to the black wire going to the light, not the cluster of other black wires; all the neutrals and grounds should be OK as they are.
While you've got the electrical box open, does the white wire connected to the black wires have black electrical tape wrapped around it? If not, it should; this will indicate that it can carry live.
What watt power transformer should I be using?
You can use a power supply like the one pictured, rated for the amount of LED's you will be driving. Read the specs on the lighting. They should tell you how many watts per a given length of the strip. So if it says, for example, that needs about 1 watt per meter, then a 15-foot section would require a 5 watt power supply. Take your total length of 76 feet, and divide by the strip's requirements. It wouldn't surprise me if you ended up somewhere in the ballpark of 20 to 30 watts.
Can I run 14 gauge wire directly from a circuit 15 amp breaker to the transformer?
No. You can't put the power supply inside the wall, and you can't bring the wire out of the wall without going through a junction box. The best way to do this is to put a cord on that power supply, and plug it into an existing outlet. There is no need whatsoever to add a new circuit or connect these directly to a circuit breaker.
Can I plug in all 5 circuits of LED strips on top of each other to the output of the transformer?
Yes, you can wire multiple strings in parallel at the power supply. If the sections are short, you can also wire them end-to-end. Double-check the specs for maximum length of a single run.
Is this the recommended way of doing what I'm trying to accomplish?
The recommended way is to plug into an existing outlet. If your lighting requirements added up to 1000 watts, you'd want a new circuit. At 20, 30 or 50 watts, it is ridiculously small in household circuit requirements.
I would also like to add a on/off switch, do they make a wall switch for 12v?
Any switch can be used for 12v. I would put the switch before the power supply. Put a cord on the power supply like this cord with switch.
Can low volt wires come right out of the wall, or do they need a junction box before being connected to the LED strip lights?
The low voltage wires don't need a junction box, but they do need a junction plate, like those used for phone jacks.
Best Answer
If you want a single control to do this, look at multi-position rotary or slide switches. It's fairly common to have a switch that connects a common to two outputs (let's call them A and B) and gives you the choices off, A only, A and B, or B only. Variants which drop one of those options are also readily available.
Switches of this sort are often seen in multi-bulb table lamps that offer several settings. They used to be common in stereo components to select which of two sets of speakers will be powered. They're also used in slightly modified form as the pull-chain switches that control ceiling fans.
A good electronics supply house should have several alternatives which meet your needs, differing in size, in how they're mounted, in the kind of knob they have (or the shaft to which you can attach a standard knob), how much voltage and current they can tolerate, and so on. You may still be able to find suitable switches at Radio Shlock, though they've phased out most of their hobbyist-friendly parts department.