... ****THAT BEING SAID, you need to decide whether or not you're going to want to take your new bird out and about with you (or outside of the house at all on a regular basis), and decide it now, BEFORE you actually bring one home, so that you can start with his/her Harness-Training immediately, from day 1; the earlier you start their Harness-Training, the more-likely they will be to fully accept allowing the Harness to be put on them, as well as accepting wearing it for a long period of time without constantly chewing on it and being bothered by it. Some parrot breeders actually start Harness-training their babies at a very young age, long before they are fully-weaned, so if you're lucky enough to find a baby bird who's breeder did this, then that's wonderful. But chances are that you'll have to start your new baby on it's Harness-training as soon as you bring him home if you want them to accept it soon, as it usually takes at least a couple of months to get them to accept putting it on and actually wearing it.
***You absolutely cannot take your bird outside without it either wearing a Harness or being in some type of cage, carrier, or backpack-carrier like a Pak-O-Bird. AND EVEN IF YOUR BIRD'S WINGS ARE CLIPPED, THEY CAN STILL FLY AWAY AND GLIDE FOR MILES AND MILES, SO CLIPPING YOUR BIRD'S WINGS IS NOT A WAY TO BE ABLE TO TAKE THEM OUTSIDE WITHOUT WEARING A HARNESS!!! Yes, I am "screaming" this at you, because I cannot begin to tell you how many people have lost their birds, usually forever, because they thought it would be fine to take them outside because their wings were clipped and they couldn't fly or gain and altitude when inside of the house...What's not obvious to a lot of new bird owners (and some long-time bird owners as well unfortunately) is that birds with clipped-wings who cannot gain any altitude inside of the house can easily fly off of your shoulder outside and take-off with the breeze/wind currents, and then suddenly they are gone, out of sight and often found 10 miles away a week or two later! I just recently helped a family in the San Diego area who had a 12 year-old son who had wanted a Cockatiel for years and who they finally purchased one for, from a breeder who hand-raises their babies, try to find their new bird...They got their son a little hand-raised, just-weaned baby Cockatiel, a beautiful female, for their son's 12th birthday, and their son fell in-love with her immediately, and had her for just 6 days when they lost her. They had taken the baby Cockatiel to an Avian Vet the day they picked her up from the breeder's home for it's very first Wellness-Exam (they had been waiting for months on a waiting-list to get a baby from this breeder for their son), and the Avian Vet had clipped the baby's wings at their request...As the father told me, they had planned on using the 2-3 months the bird would not be able to fly to work with the bird on things like Recall-Training, on going in and out of it's cage on-command, and on Harness-Training, so that once her wings had grown back-in they could let the bird fully-flighted and be able to take her out and about with them on a regular-basis on her Harness...But 6 days after bringing her home they allowed their son to take her outside into their fenced-in back yard without a Harness on and without being inside of a cage/carrier, because they mistakenly thought that since the bird's wings were clipped and she could not fly at all inside of the house, that it would be fine to take her outside...And after only a few minutes of being outside on their son's shoulder, a wild bird landed on the top of their fence while chirping; this startled their baby girl and she suddenly took-off from their son's shoulder, and to their surprise she flew up into the air and was just totally gone, out of sight. They searched for her for hours, even after dark, trying to use the "contact-calls" they had been working on in the 6 days they had her, but they could not find her...They did everything possible to find her, they followed every little bit of advice I could give them about putting "Lost Bird" ads up online all over the place, calling every local Vet office, Rescue, Shelter, Pet Store/Bird Store, etc. to inform them, and they had 1,000 color posters with the bird's photo on them printed-up and they posted them EVERYWHERE within 20 miles of their house...And they never found her. And their son was and still is totally crushed and very much traumatized by it...So that's why a clipped-bird does not equal safe to take them outside without a Harness or without being inside of a cage/carrier...
Most people like to train them to accept and wear a Harness, and the most popular and safest Parrot-Harness that most of us here and elsewhere both use and endorse is the "Aviator-Harness". It's the easiest Harness to put on them, it's very easily adjusted once on them, and it allow them to not only land-safely by slowly gliding to the ground, but it also allows them to be able to fly at the end of the included Leash while you run along side of them with the Leash attached to your wrist. So we typically recommend that you buy an Aviator Harness and start your new bird's training immediately...The Aviator Harnesses come with a DVD that has a very detailed, all-inclusive tutorial on how to get them to accept putting the Harness on and then also wearing the Harness, from the first step of just placing the new Harness in or on your bird's cage for a week or two, so that your bird becomes acquainted with it and knows that it's not of any danger to them, through each, individual step of getting them to accept putting it on, and then getting it on all at once and getting them to accept wearing it and leaving it alone while they are wearing it...It does typically take them at least a couple of months to fully-accept, but the younger they are when you start their Harness-training, the easier and quicker it will go for you...And a Green Cheek Conure wears a size "Extra Small" Aviator-Harness, so if that's what you decide on then you'll know what size to order...Actually my Green Cheek, my Cockatiel, and my Quaker all wear a size "Extra Small" Aviator Harness, as do Suns and Jendays I believe, and my Senegal Parrot can also fit into the size Extra-Small because they have so much adjustment, but he can also fit into a size "Small" too if it's adjusted to it's smallest fit...When you go to order an Aviator Harness (I think Amazon, eBay, and Chewy are the cheapest places), there is a sizing-chart that will tell you what size fits each species of Parrot...
***Also, just an FYI, as mentioned above already, when you take your bird out in-public, you just have to expect that people are going to walk-up out of nowhere and touch them, even if you tell them not to, and often times without even asking you if they can or saying a word. And you should always say "No" when asked by someone if they can touch them/pet them, because you never know how your bird is going to react, and the last thing you want is for your bird to bite a perfect stranger, especially if you decide on a Sun, Jenday, Senegal, etc. That's not to say that a Green Cheek, Cockatiel, even a Budgie or Parrotlet cannot give a really bad bite, trust me they can, especially the Green Cheeks, but a Sun/Jenday, a Senegal, etc. can take a finger right off with one bite if they want to. And unfortunately we all know that even if someone just walks-up and touches your bird without asking you if they can, if your bird bites them badly, they are likely to blame you and your bird, rather than blaming themselves. And this can get very ugly legally if your bird happens to bite them badly enough that they require stitches or even worse, if they happen to take-off a finger...And for the people who do ask you first, it's just not a good idea, even if your bird is very well-trained and doesn't bite, because #1 you can't EVER know that your bird isn't going to bite, just like your dog, and #2 it's typically too stressful for your bird in the first-place to be touched by someone they don't know...Parrots are very different than dogs, cats, rodents, etc., because they are not domesticated animals, and they bond closely with one person or one small "flock" of people they live with if you're lucky, and that's it. So it's best to just ask people to please not touch them out of their safety as well as the birds.