Have I made a terrible mistake

I'll start by saying that I'm far from an expert - I'm just speaking from my experience raising Yoshi from a 3 month old Quaker to his current age of about 1.5 years.

For training, what kind of environment are you attempting to train in? My advice is to get a training stand (which is basically just a free standing perch that will have him at least at chest height to you). Take that to an area of the house where he cannot see his cage. This should calm him down enough, and make him pay more attention to you, as it's slightly out of his comfort zone. Start off training with very small things. The first trick I taught Yoshi was to shake my hand (or, finger). I accomplished this by saying the phrase "shake my hand" then taking my finger and forcing his foot up off the perch onto the tip of my finger. He caught on to this very quickly, and within a week I could say "shake my hand" and he would lift his foot up, ready to grab onto my finger while I slowly shook my finger up and down. We only trained on this for maybe 10-15 minutes at a time, maybe once or twice a day.

I'm not 100% sure what you mean by "settle down", but I will be completely honest - when he starts going through puberty you will likely have a harder time with him. That's the point when he could start becoming cage aggressive, or protective towards certain things. I consider myself very lucky with Yoshi, he isn't cage aggressive at all. I'd like to think I played a part in that by spending time rotating toys and perches in his cage, but I can't say for sure. The main changes I've seen in Yoshi since he started puberty is that he won't let me touch or hold him in some of the same ways he would before. Before he started puberty, I could say "let me see your wing", and he would let me lift his wings up to inspect them without hesitation. Now if I try, I get bit immediately. I used to be able to say "lay on your back" and flip him over in the palm of my hand, and he would lay there without issue. I can't even attempt this now without getting bit. These are actually some of the only times he will bite, however.

As far as talking, most things Yoshi says aren't things I "trained" him to say. He picks up on things that are said a lot on his own. When he first started trying to talk (which was around 5-6 months of age, I believe) it wasn't clear at all. I realized that he would practice talking when he was by himself and (at least believed) that I wasn't around. By what I've read from other Quaker owners, this is a common thing. Now, since I work night shift and sleep during the day, I can wake up in the middle of the day and hear Yoshi in his cage chattering away. It's not uncommon for him to be saying something I've never heard him say before. And most of those things I still haven't heard him say in my presence. It's as if he thinks he has to perfect them before he will say them in my presence. So my advice there is to just give Pidge some time, and he will likely practice and start talking more clearly. Also, if you do try to train him to say phrases, make sure you say them as clear as you can.

For crying when you're not in the room, is he screaming, or is it just contact calling? Contact calls are short, loud squawks. He's doing this because he considers you his mate and wants to be in contact with you at all times. When you aren't in his line of sight, he is worried about you. He expects you to respond to his contact calls. You're best bet is to pick a sound (maybe a whistle) or a word or phrase to yell back at him when he contact calls you, and try not to use that same sound / word / phrase anytime outside of that scenario. For Yoshi, it's "pretty boy!". After he's contact called me a few times, and I've called back, he will say "pretty pretty pretty!", and then usually stop. At least for a minute.

Parrots are a handful, and can be challenging. They definitely aren't for everyone, and they rarely are that perfect picture you have in your mind after watching the cute video on YouTube. That's not to say that every single one of them can't have moments of being that perfect picture, but there is a LOT more to them than that.

My advice is to stick with it. Do your research, read posts here, ask questions - we are here to help you in any way we can.

Great advice from zim and biteybird. As biteybird said, Pidge is still pretty young. In the wild, he would still be under the instruction of adults, and you have to assume that role. It's our responsibility to help them adapt to the very unnatural environment we place them in. It's a lifelong responsibility, and it can feel overwhelming at times, but it's also very rewarding once you get things on track and stick with it.

Parrots chew, and that is so fundamental that you won't be able to change that. Make sure he has lots of fun things of different textures that he can chew on. Reward him when he chews on his own stuff. If Pidge is free flying, you have to bird-proof as best you can the areas he has access to. Limit the areas he can access so that you can always keep an eye on him, and if he starts chomping on the furniture, distract him with one of his toys, and reward him when he plays with it.

Make sure all your interactions with him are calm and positive. Parrots easily pick up on our emotions and moods, and will reflect them back onto us. If you're feeling despair over the lack of progress with Pidge, leave him alone and wait until you can put on a happy face. Keep at the training. Break a desired behavior down into tiny increments, and reward lavishly for the smallest bit of progress.

The best rule when dealing with parrots is "reward the behavior you want and ignore the behavior you don't want". Obviously, you can't literally ignore Pidge while he shreds your prized possessions, but you can calmly redirect him. No drama when he misbehaves - that is a reward for bad behavior, even if you don't see it as a reward.

You care enough to ask for help, and that is something many people won't do. They'll just give up and put the bird out of the way and go on about their lives. It takes work, but you can turn things around! Keep coming back for help, or when you just need a little understanding.

Thank you everyone for the great advice!
I'd like to know how long most people leave their bird in a cage. Pidge comes out when he wakes. We have a shower (best place for conversations) and then breakfast. he has his own food but is obsessed with sharing my bran cereal. I leave him out all morning. In the middle of the day he has children's television and he goes back in his cage. He practises his vocals all afternoon until around 4pm then wants out again. He's out until we start cooking. By 6:30 he's slowed right down and then at 7:30 when he goes to bed he throws himself around the cage and is very noisy. I give him a scratchy cuddle and into bed at 7:45.
He now clucks and gives me a voluntary kiss and can shake hands.

Pidge sounds like she is coming along just fine. Oscar is always out of the cage as he does not fly and I think that makes a big difference. Also because we are both retired we can keep an eye on him at all times. We do go out to go shopping etc... and Oscar will still be out of his cage ... usually sits on top and waits for us to return. Everyone is a little different and the Quakers usually adjust to the schedule you set for them.

It seems you are letting her out enough but I do understand why she would put up a fuss and want out while you are there. These are flock birds... and they always want to be with their flock.

It depends on the day. Casey gets wakes up around 7 during the week, but doesn't get let out of her cage for breakfast until 9. She is usually out most of the morning she prob goes back in around one. Then comes back out for dinner around 6. If we are home in the evening her cage is left open, sometimes she is on top but she will go back inside too. Around 8 each night she has a loud long conversation with herself, repeating all the things she has heard in the day. She gets very animated. Between 9 and 10 she starts calling night night night until I come over to clean her cage and get her ready for bed