Destiny 2 will finally let you change your character's appearance, with one big limitation
As a jaded veteran, I can't say I'm surprised it'll lack the biggest option
If you want to change your Destiny 2 wizard's appearance, the only current method is to stash your wizard's gear in your vault, create a whole new wizard with a new look (after deleting the old to free a character slot, if needed), grab your old wizard's gear, then optionally replay any story bits to catch up. The MMOFPS has no barber, no surgeon, it's new character or nothing. After six years (longer, if you count the first game), Bungie are finally planning to change this. An upcoming update will introduce the ability to change your character's look—within limits. For drastic changes, you would still need a new wizard.
In the latest weekly This Week At Bungie blogblast, the developers explained this feature will arrive in an update before the launch of The Final Shape. That expansion, which will close out the ten-year 'Light and Darkness Saga' story, is due to arrive in June. The option will be available from the character select screen.
"You will be able to modify the appearance of your Guardians anytime, without cost or limit to the number of changes," Bungie say. "Just take into consideration that while you can change the body type, face, hair/head, and markings, it won't be possible to change from one origin to another (i.e., Human, Exo, or Awoken)."
That's a pretty big caveat. While swapping haircuts and makeup and body type (previously referred to as male/female and masculine/feminine) and antennae and tattoos and mouth glow and such is extremely welcome, it's a shame you can't change your origin. That's the most dramatic part of a characters look. Exos are people transferred into cool robot bodies, Awoken are humans gone blue and glowy after millenia in a high-speed pocket universe, and humans are just those bipeds you see around you down the Tesco. They all look strikingly different and, while our wizards rarely speak, they do have different voices.
I've heard of people recreating characters after years to change their look. I will confess I myself did recreate my Titan after only a few months of play to change a haircut I grew to hate, because Destiny's true endgame is fashion and I play to win. I've also known people to always have their helmet shown, to hide their stupid face. You don't lose a huge amount by recreating characters, as long-term progress is tied mostly to items and those are interchangeable, but you will have to replay per-character story parts and quests if you want to be up-to-date (and to not have quest logs and markers bugging you). It'll be nice to not go through the rigmarole just to cut your hair.
Of course—I say with all the weariness of a Destiny player who quit cold after thousands of hours because I was sick of Bungie's natural instinct to design every game system in a miserable, miserly, and grindy way—you would expect Bungie's first attempt at a long-desired feature to be incomplete and a little frustrating. Give them a few years and maybe you'll get to change origin too.