What does the Moon's orbit around the Sun look like? I'm curious as to what the Moon's orbit around the Sun looks like. If there's an answer, what's the intuition for it? Here are some things I'm assuming when trying to tackle this question:


*

*The Moon's orbit must be concave toward the Sun.

*The Moon speeds up as it goes toward the Sun, and it slows down as it moves away.

*For an observer on the Earth, the Moon appears to orbit the Earth roughly $13$ times a year.

*The Earth, the Moon, and the Sun remain in the same plane.
 A: Incorrect Path

I'm curious as to what does the moon's orbit around the sun looks like?

One might think the orbit (in the sun's rest frame) follows the path of an epitrochoid.
A (very) over exaggerated view of this motion (for unrealistic parameters, thus, not an accurate representation) can be seen in the following animation:

Note that if you change the ratio of the different radii to values that are to scale, then the plot would look more like an epicycloid as in the following example animation.  The orbit is more realistic but still exaggerated because it would be impossible to show half an orbit to scale.

The correct result is shown in the zoomed-in view of David Hammen's post above.
Update/Correction
The above animations are flawed because the little red dot orbits as fast as the blue circle rotates as it "rolls" around the large red circle with no slippage.  For a realistic Earth-moon system, there should be a lag between the rotation of the blue circle and the red dot, as if the blue circle were "slipping."  Or equivalently, one would not use a rigid axis connecting the center of the blue circle and the red dot.  This would result in there never being a negative velocity of the moon relative to the sun in the fashion shown by the epicycloid path in the 2nd animation above.
Incorrect (still exaggerated) Path
The correct path uses two different orbit rates, one for the Earth about the sun (i.e., 1 year) and one for the Moon about the Earth (i.e., ~27 days).  In the following example, which is still an incorrect (i.e., exaggerated) orbital motion but much better approximation, I exaggerated the ratio of the astronomical unit to Earth radius by a factor of 100 and increased the moon's orbit by a factor of four to help make the visualization more obvious.

Correct Path
The following example does not exaggerate the orbital periods relative to each other and is a zoom-in of the above graphic (oddly the GIF, created through the same methods, does not loop on my screen).  Here the orbit is always convex with respect to the sun, as David has correctly stated.

A: 

*

*The Moon's orbit must be concave toward the Sun.


The Moon's orbit with respect to the Sun is always convex. This is easily proven by comparing the minimum possible gravitational acceleration of the Moon toward the Sun (5.7 mm/s2) and the maximum possible gravitational acceleration of the Moon toward the Earth (3.1 mm/s2). The acceleration vector, and hence the curvature, is always inward.
This means the Moon's orbit about the Sun doesn't look like either of the two images below:

(Source: https://blogs.discovermagazine.com/badastronomy/2008/09/29/the-moon-that-went-up-a-hill-but-came-down-a-planet/#.V3xtsDcgtOo)
Phil Plait is normally very good. This is one of those places where he was bad. The Moon's orbit about the Sun instead looks like this:

(Source: http://www.math.nus.edu.sg/aslaksen/teaching/convex.html)
A close-up view:

(Source: http://www.wired.com/2012/12/does-the-moon-orbit-the-sun-or-the-earth/)



*The Moon speeds up as it goes toward the Sun, and it slows down as it moves away.


Yes and no. The “yes” part is that the dominant feature of the Moon's orbit about the Sun is that the Moon orbits the Sun with the Earth. Ignoring the Moon's acceleration toward the Earth, the Moon's acceleration toward the Sun is greatest now in early July when the Earth/Moon system is closest to the Sun than at any other time.
The “no” part: The Moon is moving slightly away from the Sun when the Moon is in the second and third quarters. This is when the acceleration toward the Earth coincides with the acceleration toward the Sun. In the fourth and first quarters, the acceleration toward the Earth is directed against the acceleration toward the Sun, slowing the Moon down a bit with respect to the Sun.

For an observer on the Earth, the Moon appears to orbit the Earth roughly 13 times a year.

This is correct. More on this below.

The Earth, the Moon, and the Sun remain in the same plane.

This is incorrect. As noted by Diracology in his answer, the Moon's orbit about the Earth is slightly inclined with respect to the Earth's orbit about the Sun.

Note that one of the sites I referenced (http://www.wired.com/2012/12/does-the-moon-orbit-the-sun-or-the-earth/) asked whether the Moon orbits the Earth or the Sun. That blog erroneously came up with the answer that the Moon orbits the Sun rather than the Earth. Looking at acceleration or the shape of the orbit is the wrong way to look at the question “Does object A orbit object X or object Y?”
One answer to the question of “does the Moon orbit the Earth or the Sun?” is “Yes!” "Orbit" is not a mutually exclusive term. If one insists on a singular answer, the right way to look at things is via energy rather than acceleration. Since the Moon is gravitationally bound to the Earth, the Moon does indeed orbit the Earth.
A: Before answering let me mention that there is a terrific free applet showing the orbits, including the velocity vectors of the system Sun/Earth/Moon:
https://phet.colorado.edu/en/simulation/gravity-and-orbits
It is in java so pretty easy to download and use.

The moon's orbit must be concave toward the sun.

The Moon' orbit around the Sun is a composition of the Moon'orbit around the center of mass of the system Earth/Moon and the orbit of this center of mass around the Sun.

The moon speeds up as it goes toward the sun, and it slows down as it
  moves away.

Actually is the other way around (in the Sun's frame of reference). The moon slows down when it moves towards the Sun and speeds up when it moves away. You can see this by doing a velocity composition. When The bodies are ordered as Sun--Earth--Moon the velocities of the Earth (approximately the velocity of the center of mass of Earth/Moon) and the velocity of the Moon are approximately parallel and the resultant is a large vector. When they are ordered as Sun--Moon--Earth, the velocities are anti-parallel and the resulting one is a small vector.

For an observer on the earth, the moon appears to orbit the earth,
  roughly 13 times a year.

Right.

The Earth, the moon, and the sun remain in the same plane.

No. The plane of the Moon's orbit is slightly different than the plane of the Earth's orbit. The difference is about $5^\circ~.$ That is why not every new and full moon is an eclipse (solar and moon eclipses, respectively). Only twice a month (28 day actually) the moon intersects the plane of the Earth's orbit. If this coincides with the alignment Sun--Moon--Earth we have a solar eclipse. 
A: The orbital speed of the earth around the sun is about 30 km/s, whereas the orbital speed of the moon around the earth is about 1 km/s.
From this it follows that at no point of its path around the sun the moon will ever show a backwards motion.
The path is similar to the trajectory of a point (moon) on the perimeter of a (somewhat sliding) wheel rolling around a large circle.
