I have a card that has three bands: black, gray, and white. If I place this in a photograph I am taking that I want to be color accurate, will that give me sufficient information to tune the photo, such as with adjust levels in Photoshop CS6, to get a color accurate photo? I know that I can adjust both color temperature (white balance) and tint, so perhaps this only gives me one dimension of accuracy (e.g. temperature) and I need a color card for the other (e.g. tint)? If true, what kind of color card would I want to get to ensure accurate color reproduction? For example, I see this one on Amazon, is that the right kind of card to get accurate color reproduction? Note: I'm taking these photos with a Nikon D800.
For normal lighting, yes.
For bizzare or novel light, no.
"Normal" means sunlight or incandescent lighting, or other lights that try to immitate that (since it's what our eyes work with). Poor "color rendering index" lamps will be lacking but that's how it looked in person too!
For odd colored lights, you need more data points to know just what is happening, and you still have underlying ambiguity: correct for blue lights and you don't know if the object was supposed to be blue or was grey.
The color is subjective in photography, since we're not making scientific instrument readings. So you end up adjusting the actual green leaves in the picture to match the impression of the live scene, and don't need a formal green folage patch on a color bar.
A color chart can be handy for matching across different shoots, calebrating the camera profile, and learning about the camera's response; and of course in understanding "nasty" lights even if you don't rely on automatic correction based on them.
I found the x-rite "passport" useful for those reasons. And I could still pull it out if I wanted a reference for (e.g.) heavily-filtered stage lighting.
Grey charts are for accurate white balance. Accurate colour depends not just on the white balance, but also on the quality of the colour transform (colour profile) and its suitability for particular light in the scene.
White balance is often a good start, but it does not solve it all, far from it. If you are not getting acceptable colour with just white balance, you can try using a different stock profile, or a different raw converter, or, if everything else fails, profile your camera. The most user-friendly target for that currently is ColorChecker Passport.
All you need for White Balance corrections the easy way is just a good neutral card, and a good White Balance tool. You already have them. You click the card in a test photo (card in the same light with the subject), which tells the computer "This spot is neutral, make it be neutral". And it does, it makes the neutral color be actually neutral, i.e., no color cast in the image. Couldn't be better. Very easy to apply to your final picture(s) too, same click (assuming all are in the same light).
The sets of three cards are never stellar quality, and its white card is likely better (more neutral) than your gray card (18% gray is not color controlled, and it's really too dark for WB anyway). A WhiBal brand card is good, but more expensive. A 5x7" Porta Brace White Balance card is very good and inexpensive, $5 at B&H... all you need. It is nothing fancy, just a known neutral color. Your white card is probably OK too.
Your Photoshop Levels tool has the center Gray Picker which does this, sort of marginally, OK for mild jobs (but it does not do multiple images). See the Adobe Help for Levels. But Adobe Camera Raw (ACR, which comes with Photoshop CS6 or Lightroom, or Elements too... all three are the same good tool) is a much better WB tool than Levels, good for the hard jobs too, with much more range and it does multiples. ACR is Raw software, but it works great with JPG too (but if you're going to use it, it might as well be Raw to get all of its benefit).
You get into ACR from Bridge, select one or more or all images in Bridge, then RIGHT click on them and use dropdown menu Open In Camera Raw .... including JPG files). (You just have to realize it does lossless edits, meaning, you must output a new JPG for other software to see the edit. Photoshop menu File - Scripts - Image Processor is one good way, it is also a batch way to do this).
This WB tool does both Temperature and Tint in one click. It simply makes the clicked spot be neutral, no color cast. Since you click a known neutral color, it works perfect. It can do multiply selected photos in the same one click (on the test picture with the card), so long as they are all in the same light.
See http://www.scantips.com/lights/whitebalance.html for more, including about the Photoshop Levels Gray Picker, or better, also about ACR WB.
Raw is sort of a different philosophy. http://www.scantips.com/lights/shootraw.html