Both are under Profile → Your details and sign-in details.
Changing your password. Enter your new password twice. You stay signed in.
Changing your email address. This is also your username. You get a confirmation email at the new address, and your address only changes when you tap the link in it. Open that email on the same phone, and don’t sign out before it’s done.
Signed in with Apple or Google? Then you manage your password with Apple or Google, not with us.
Deleting your account. See Deleting your account.