The previous answers are well written and describe the process well. I can only add my personal experience to this which, of course, may vary from one person to the next.
I find that both the "bash it out with a mortise chisel" and the "drill it out and clean up the edges" methods work about equally well and are both a lot of work. Not that work is a bad thing; but the process will take some time. Using a mortise chisel had the benefit for me of ensuring that the walls of my mortise were parallel.
As to the tenon, I found it tricky to get the sides and shoulders perfect using only a chisel and ended up buying a shoulder plane. In fact, I bought this one [from Lee Valley.] This made it much easier to get the tenon sides and shoulders to fit well. For some reason, these shoulder planes are always expensive; I was unable to find one on Ebay, or any of the online tool dealers that was both in good condition and less than $100 USD. So I just bought a new one and the one from Lee Valley is very nice to use. Most importantly, it is easy to adjust.