Matt,
I wanted to add to the great advice that is already posted here. First and foremost, birds do need time to adjust to a new area. I think that this can really apply to any animal! A cat in a new home would need time to learn it's way around the house and who the family is. Just keep giving it time.
I also wanted to share some information that applies to most parrots. You said in your first post that Bellow is your first bird. This is not an all inclusive list, but it should get bellow's and your relationship off to a good start. I would really suggest that you keep reading, here and other sources. You can always stand to learn new information, even more so when it effects the life of animals in your care.
Sleep:
Birds love sleep! They really do. You need to make sure that Bellow is getting 10-12 hours of sleep a day. You should also get a cage cover, if you do not have one yet. When it is "sleepy time" you should cover the cage. I like to cover three sides of the cage and leave the side against the wall open. You can use almost anything to cover the cage, as long as the material is safe. If in doubt, they do sell covers that are designed just for bird cages.
Make sure that if Bellow is in your room, you have the lights out and try to keep the noise to a minimal. I have two budgies right now and they stay in my room. When the sun goes down, I don't let anyone talk in my room.
Diet:
A healthy diet will really help keep your bird in tip top shape. You should aim for a mostly pellet diet with a small amount of seed every day. You can and should add a healthy serving of fresh veggies and fruits. Some suggestions to get you started are Kale, Broccoli, jalapenos, red bell peppers, apples, oranges and bananas.
Fresh foods should be out for no more than an hour, so don't give to much at once. If Bellow is not already on a pellet diet, you need to slowly change him or her over. For most parrots, just putting pellets in there will not work. You need to mix and match with seeds. A good starting area is 50/50 diet of seeds and pellets as a base. You want to keep increasing your pellets until it is 80-90% pellets and the rest is seeds. The idea here would be to have pellets in the cage all day, changing them out for fresh ones each day. The seeds will usually be eaten fairly quick so should be gone with out needing to throw them away.
Water can grow bacteria very quickly! If Bellow has water "dishes", I am certain he or she will get them messy as well. If you are using water dishes you should change the water twice a day. Fresh water goes a long way! You should also use distilled water. Tap water has tons of added chemicals, including chlorine. If you have a bottle with a drip proof end, you can get away with changing it once a day. Make sure that you clean the bottle and end good every day.
Water and food dishes should be wiped clean. I've already mentioned above about changing the water, but cleaning the dish is very important. When you change the water, make sure to give it a good cleaning with a rag. You should clean your food dishes every day as well.
Cage:
I am not sure what the spacing of your cage bars are. This is a very important thing, because it can have some bad effects. If the bars are to wide, your feathered friend can get his or head stuck. If the bars are to narrow, feathers and nails can get stuck. For a GCC I would not go above 5/8ths inch. Your mileage may very here depending on how big or small Bellow actually is.
You need to also make sure that there are plenty of toys and perches in the cage. A bored parrot is an unhappy parrot. For a single GCC you should have at a minimum three to four toys in the cage. Make sure to clean the cage bars, perches, and toys every week. You should spot clean any droppings every day. Make sure to change out the toys as they are chewed up. Conures in general LOVE to chew toys! When the toys get chewed up to bad, they could be dangerous. Parrots can also get bored of toys after having them for to long. If you notice that Bellow is no longer playing with a toy, you could try retiring it for a bit. I also suggest adding random foods that Bellow likes to toys on occasion. This will allow Bellow to forage here and there, increasing happiness.
If you have wooden dowels in your cage (You know, the long wooden perch that goes from one end to the other with the same perfect smoothness all around) you should take them out. These can damage birds over time, causing all sorts of problems. They also discourage exercise in the cage. You need to have a good mix of different size perches, spread around the cage. Make sure that most, if not all, perches do not go the entire way across the cage. This way, bellow has to move around the cage more then just going across the perch. You need to look into getting perches that have varying widths on the perch itself. Most pet stores that sell avian products should carry these. I would also suggest a rope perch. I've had many birds that fall in love with these!
Bedding is the last area I will talk about here. Be careful with the bedding you use. Some parrots will actually eat the pine or straw bedding, and this can be harmful. Most people use newspaper or printer paper. I really like the printer paper because it is clean white paper. This allows me to see the droppings and judge any changes in my birds.
Other non specific advice:
These will just be small quick points. These are things that I've seen posted about tons here or have heard of friends having problems with in the past.
You shouldn't give Bellow any more affection now then you can afford to give the rest of his or her life. This means up to 30 years, depending on age! You said in a post that you have the cage open and Bellow out most of the day. Be prepared for the day to come that you can't give this and Bellow gets very upset with you. This could even result in serious, life threatening problems such as plucking and mutilation. Please, be careful here!
You can expect at some point or another to have some loud screeching! Don't go running into the room right away. This is actually a call that parrots will do to find their lost flock members. First, I would say be glad that you are hearing this! That means that Bellow is a proud member of the family. Second, don't go running to it. When you do this, it reinforces the behavior. Stand outside of the room and wait a few minutes for the screeching to stop. Then, go into the room to show that you will come in for the more acceptable silence.
Parrots need lots of activity during the day! This stimulates their brain and relives boredom. In captivity, we can give our friends this through small training sessions. You can google search for all sorts of training ideas. Just make sure that training last no longer than 10 to 15 minutes.
Parrots do NOT like to show that they are sick. This is VERY important to understand. As a member of the flock, a sick bird is seen as danger. They can slow down the flock and even get others sick. Parrots will do everything in their power to pretend that they are not sick. You need to understand this and pay attention to all of the small details. Some signs that you should look for are nasal drainage, strange or overly watery droppings, unusual behavior, lack of interest in toys and not eating. There are many more signs to look for, but this should get you started. If you suspect that something is wrong, a vet check should be done. A little money spent for the welfare of your bird is much better then Bellow's death.
I apologize for the super long list of information. I wanted to present you with things that I felt was very important for a first time ever "parront". Your post show that you are extremely interested, and I hope that you can find some good information here. Like I said at the start of this, I encourage you to keep reading. There is an almost limitless supply of information available out there. I wish you and Bellow the best of luck.