big bang theory - How do we know the universe's expansion is speeding up?

As a fellow layman, I'll give this my best shot.

The further away a galaxy is from us, the faster it is moving away
from us. But the galaxies we see exist in the distant past because it
takes a long time for their light to reach us. The closer galaxies
are, the more recent their light is, and the slower they move away
from us.

All true. Think of a single point explosion, at any given time, the object twice as far should be moving twice as fast, cause they both started in the same place at the same time, so twice as fast moves twice s far. There are, complications - with the explosion there's air resistance, with the universe well, I'll get to that.

Doesn’t this seem to indicate that the expansion of the universe is
slowing down, because the more recent light from galaxies, appears to
be moving away slower? Does this make sense?

OK, it sounds like you have an incorrect assumption there. Expansion means the far away is moving away faster and the closer is moving away slower, that's true with accelerating or decelerating expansion.

What they look for when studying expansion is measuring the precise speed and comparing that to distance. In steady expansion, you expect to see the galaxy 8 billion light years away to be apparently moving away twice as fast as the galaxy 4 billion light years away. (I say apparently cause we can only observe the relative velocity from 8 billion years in the past).

So, whether the galaxy 8 billion light years is traveling slightly more or slightly less than twice as fast away from us as the galaxy 4 billion light years away - that's what tells us about acceleration vs deceleration.

Imagine a car driving 60 MPH, and the driver puts the ever so slightest pressure on one of the pedals, but you don't know which one, and similarly the car driving 30 MPH does the same, but we can only see how fast the 60 MPH car was driving 6 minutes ago and we see how fast the car driving 30 was driving 3 minutes ago - that gives the 30 MPH car more time to apply the acceleration or deceleration. It's that time differential that gives us the information.

So, like the cars above, the galaxy 4 billion light years away has been in the accelerating expansion of space longer than the one 8 billion years ago and that's what tells us that the acceleration of expansion is happening, the nearer galaxies are moving a bit faster away from us than they would in a steady expansion or a gravitationally slowed expansion. (edited to make a bit more clear)

The newer light is moving slower. The older light moving faster. So
from this it seems that the universe was expanding faster in the past?

light doesn't move slower or faster. it red-shifts when objects are moving away from each other. The faster they move away the greater the red-shift. That's one way relative velocity can be measured.