Is there a Lagrange point between the earth and moon where a space station could sit forever without orbiting around either? Just curious, but it seems like a place like that would be perfect for either a second space station or maybe like a giant tv that could display international weather or something. Or maybe a giant laser that can shoot down all that space debris.
Yes. The Earth-Moon system has a Lagrange point L1, positioned between the Earth and the Moon, It is about 85% of the distance to the moon (about 320000km compared to 380000km.) A body at L1 would orbit the Earth, once every month (it would be in a 1:1 resonance with the moon)
L1 is an unstable point, so you would need to use rockets to keep the satellite close to the L1 point for an extended period of time.
320000km is much further than the geostationary orbits, (at about 36000km) and much much further than the Low Earth Orbit of the ISS (about 400km)
There are practical reasons not to put a space station there: It is much harder to get to, with few practical benefits. It is too far for "tv" or "lasers".
@JamesK's answer is good, but I'll address some of your points more specifically.
Is there a Lagrange point between the Earth and Moon...
Yes, it is called Earth-Moon L1 (to distinguish it from Sun-Earth L1)
...where a space station could sit forever...
As pointed out, you could put a station there but you'd have to do some station-keeping maneuvers using propulsion of some kind to keep it there.
...without orbiting around either?
It's always a little tricky when we talk about what does or doesn't orbit. You can say the Moon orbits the Earth, but really they both orbit the center of mass of the Earth-Moon system.
And both the Earth and Moon orbit the Sun.
But let's just say that the Moon orbits the Earth, and the five Earth-Moon Lagrange points orbit along with the Moon once a month as well.
You can watch the Lagrange points orbit Earth in the really cool 3D4U video Earth/Moon Lagrange points animation which should also be viewable below, and was found in https://space.stackexchange.com/q/14337/12102
Basically we usually say that the Moon and the five Lagrange points orbit the much more massive Earth, but mathematically they're all orbiting the center of mass.
Yes, the Earth-Moon system has Lagrange points, and Wikipedia has a short list of satellites that have (or will) orbited near L2, L4 or L5. Neither L1 nor L3 appear to be popular.
The Moon's orbit around the Earth-Moon barycentre has an eccentricity of ~0.0549 and it's quite dynamic, with relatively short apsidal and nodal precession cycles, primarily due to perturbation by the Sun. Alternatively, the Moon's orbit around the Sun is strongly perturbed by the Earth. ;)
The mean Earth-Moon distance varies over the year, as I mentioned here. It's interesting to compare that distance with the distances to the Lagrange point. Here are the relevant graphs:
The mean distances are
All of those plots were created using the script in the linked answer. The Horizons ID for the Earth-Moon L1 point is 3011.