For Android Pay, assuming you have the compatible watch, you can either assign a side button to Pay or you can launch it from a complication. When you tap it, it will bring up Pay, and if you have more than one card on your account you'll need to select the one you want to use.
Otherwise, you just hold your wrist to the terminal, make sure the top edge of the watch is properly aligned with the NFC point, and wait for confirmation. It's straightforward in theory, though sometimes aligning the two perfectly can be tricky in practice.
With Apple Pay, it's a case of double tapping the side button and then holding the Watch up to the reader. Pretty similar.
It's here where Samsung Pay has an edge though, as it's the only one of the three to support MST (magnetic strip transactions) as well as NFC. This means it will work at almost all terminals with a magnetic strip reader, although the process means tapping your watch against that bit of the terminal instead. This is also a tad fiddly, but worth it for the added support.
For us, Samsung wins the battle of usage on smartwatches purely down to having both NFC and MST support, while the other two only support NFC. The fact it also works with Android phones makes it even better, and iOS support would be even more fantastic.