Is velocity relative?

Your intuition only works for speeds on the scale we're used to.

If you think you're standing still while I drive past you at 60 m/s, then I think I'm sitting still and you're racing past me at 60 m/s, and meanwhile a slower car thinks I'm moving forward at 30 m/s and you're moving backwards at 30 m/s.

"Velocity is relative" means there is no objective reference point. The opinion that we're moving at +30 and -30 m/s is equally as valid as the one that says you're standing still. Your reference frame isn't special just because the road and trees are moving at your speed. All the laws of physics work exactly the same if we assume I'm the stationary one.

Here's where it gets weird. Observers moving at different speeds measure all light to be moving at one constant speed. If I race along at 1/2 c (relative to you), then I do see you moving away from me at 1/2 c, but I still see the light moving at c.

"Velocity is relative" does NOT mean that the relationship is specifically v3 = v1 + v2.
Einstein and Lorentz found that it's actually v3 = (v1 + v2) / (1 + v1 v2 / c^2).
As long as v1 and v2 are small compared to c, this approximates to v3 = v1 + v2. But try it with v2 = c and you get out v3 = c.

The invariance of this speed of light has consequences for how we measure distance and time.

Not quite...

Light's speed is always within a frame of reference. Within that frame of reference, it is always the same (in vacuum). It's a scalar quantity, with magnitude (measurable size) but independent of direction
Velocity depends on direction as well as magnitude. So, it can have a specific speed, but depending on direction, it can be quite different.

Now, in terms of speed: There are galaxies that seem to be moving away from us at more than the speed of light, so, switching things around, we're moving from them at more than the speed of light. But our velocity is never more than the speed of light.

It's hard to get your head around the difference (and it is for most people, even those studying the subject), but until we can start getting close to light speed, we'll never be able to really show people how to grasp it intuitively.

"But if velocity is relative, doesn't that mean that given a certain reference point, both light and I are traveling at half light speed?"

For all but one frame of reference, you are not traveling at 1/2 c.
For all valid frames of reference (which means anything just shy of c), light always travels at c.

So no.

Light itself has no frame of reference (light can never see light leave at c).

Now you might insist that "well then velocity isn't relative". To which I'd point out that Nature refuses to answer the question of what One Way Light Speed is, and *any* attempt to measure the speed of light must return a Two Way Light Speed measurement. So even the speed of light over one path might be relative, but the sum over the path twice (there and back again) is not.