When you train up to level 20 in City of Heroes, you earn yourself a free makeover at Icon, the tailors to the heroes. There's a mission to undertake for Serge, the manager, and you should get to that. Afterwards, however, you're eligible to earn your cape. Problem is, you're not really told that or who to see, and earning it at level 20 is a darn sight easier than waiting until later on.
You don't have to wear one right away, certainly, but if you want the option, it's best to do earn your eligibility sooner rather than later. Later it will be much harder to accomplish, and possibly impractical to do solo.
Go to the City Hall in Atlas Park and enter by the main doors. Right in front of you will be the city representative, Paco Sanchez. Speak with her and she will give you the basic mission. You will visit three people in three zones who will tell you of their personal experiences with Hero 1 of the lost Omega Team. After you have spoken with each of them, you will be sent to the memorial in an abandoned Rikti base where Omega Team's time-capsule resides.
Lay the beat down and exude justice from your very pores. Clear every villain and minion out of the place. Note that the elevators are less than obvious, so check your map for the green markers representing them. On an upper level you will find a number of foes around the time-capsule. You must not let them destroy it! If you're closer to Level 20, you'll find the foes aren't really all that interested in smashing the time-capsule, and your regular combat tactics should carry the day.
Return to Paco Sanchez after your victory has earned you a healthy slice of experience and prestige, and capes will be unlocked for you. There's no need to run out and get one right away, but by all means visit the tailor and at least look over the options. There are a lot of styles and ways to customize it - you may just find the perfect thing to offset your costume.