The speed of comets can vary a lot, depending on its orbit and where it is in it. The closer they are to the Sun though, the faster they’re going. We think the fastest ever comet went at 366 miles a second back in 1843.
The tail of a comet points away from the Sun because it’s pushed by radiation pressure (basically light pushes it back) as well as the solar wind (hot gas that streams out of the Sun). Without these, the comets would have a blobby shaped atmosphere of gas, water vapour and dust around them.
The fastest a comet can go depends a bit on where it came from, the fastest ones come from the “Oort cloud” because they’ve come to us from a long long way out and have been accelerating all the way. Their fastest speeds are close to the Sun’s escape speed, which is a few hundred kilometres per second! (the Earth goes around the Sun about ten times slower than this).
If you google images “comet tail” you’ll see that they have two tails, one from dust and one from gas (the gas one is bluer). It’s the gas one that points directly away from the Sun because it follows magnetic field lines and is pushed away by the Solar wind. The dust one points nearly directly away from the Sun because it’s pushed away light from the Sun, but not as strongly.