There is a huge amount of information on this topic. Start with "Hubble Law".
There is even the CLEA exercise that gives you a chance to go through the measurement steps.
As an outline to hang your own reading:
When you take the spectrum of a galaxy it has narrow dips in brightness (=lines) at several wavelengths. These come from the cumulative spectra of all the stars in the galaxy. If a star or galaxy is moving toward or away from us the wavelength of these lines is shifted toward the blue (higher frequency) or red (lower frequency) end of the spectrum (see Doppler Effect). The faster the speed toward/away the bigger the shift in wavelength/velocity.
The analogy is with the pitch of a train whistle (or car engine) as the train approaches the pitch (frequency) is higher and lower as the train recedes. This is respect to if the whistle was standing still. It is the relative motion, so if you and the train were traveling along side by side you would not hear the same pitch as if both were standing still.
Starting in 1940's the spectra of galaxies have been measured. Only a few of the closest galaxies were found to be moving toward our galaxy. All the rest in every direction are moving away. What is more, the further the galaxy is away, the faster the speed of recession. There are a number of complementary ways of measuring distances to galaxies, which now agrees with each other. (look up cosmic distance ladder, or Chepied variable stars)
If it was just us moving then galaxies in front o us would be moving toward us and galaxies behind us would be moving away and we wouldn't see the pattern of further galaxies moving away faster. The other obvious idea is that our galaxy is at the center of the universe and everything is rushing away from the Milky Way. Cosmic BO or explosion. But looking around we don't see anything special about our galaxy. We can explain the measurements by having the whole universe expanding in all directions at once.
A 3D analogy is a raisin cake with each raisin being a galaxy, and the dough being the space of the universe (cake). As the cake expands the raisins are dragged along, all moving away from each other. There is also the 4D analogy with a balloon being blown up.
A neat little experiment. Take a sheet of paper, draw little circles representing galaxies randomly on it. Photocopy this drawing onto a clear overhead sheet, but slightly enlarges say 125%. Then overlap the overhead with your original centering it on a "galaxy" near the middle. You can see all the other "galaxies" are moved away from the central one. The further away the greater the shift. If the difference between the two was in time, the shifts could be converted to speeds. ie the further from the center the larger the velocity. You could even make a plot of velocity (shift) versus distance from the central galaxy (origin). Now pick another "galaxy" and center the overhead on it ! You'll see the same pattern of shifts but now with respect to the new origin.
The last part of the original question was about the universe accelerating.
Here's is a diagram taken from
http://www.valdostamuseum.org/hamsmith/MRRsnIaSZgl.gif;
Instead of being a nice straight line the measurements follow a curve, with a lower slope (flatter) toward the right (larger distance). Further away means further back in time. The slope of the line gives the speed of expansion. So a lower (shallower) slope means a lower speed. So between then and now the speed has increased, so the rate of expansion has increased ie the universe has accelerated. (look up type 1a supernovae).