Obviously the poor little guy needs to get out of there and start getting showered with attention... but - and please be very specific - HOW do I start this process of getting him handleable and eating a more varied diet?
READ THE THREADS LINKED ABOVE! Then read them again and again as you start the process below! Good reading out loud material for building trust with your bird!
You start by mentally preparing yourself for a long, slow and difficult journey with this bird. Have no expectations out of him to be any certain way, be willing to work at HIS pace and be sure you're in it because you genuinely want to and feel able to help. PATIENCE and understanding, you'll be needing LOTS of those too

It took about a year and a half before our rescue amazon began stepping up reliably without biting and several more years before he started going out and about and socializing with other people. The effort and bites early on was difficult, painful and hard to keep the focus on the end result of a happy, healthy, well adjusted bird. He still doesn't like to be petted but he does love being around us and is just such a happy little creature these days. The result at the end of the day has been 1000% worth the effort/pain initially. He had a rightful distrust of humans and expected/required us to earn his trust and friendship, the hard way. You MUST expect and accept those terms and conditions as a distinct possibility with a bird from a situation like what you're describing.
So now you have the bird home- I would suggest getting the initial vet visit out of the way ASAP in case any medications need to be administered (which can definitely break any gains in trust you build early on). Our bird needed 2 rounds of antibiotics and a few follow up vet visits immediately after being adopted. Lets just say that was a less than fun experience for everyone involved. Once vet checked and healthy, THEN start trust building. The best 'tried and true' method will be sitting next to the birds cage several times a day and just reading or speaking to him in a soft tone of voice, getting him used to you as a non-threatening presence. Every member of the household should be doing so too. Once he seems more comfortable, start bringing bird friendly snacks to these sessions, whatever treat he likes best. Eat or pretend to eat the treats yourself, making a big deal about how yummy they are. Amazons are winged pigs and undoubtedly distrust will eventually give way to desire for treats and he will approach. Gently offer the treat through the bars and praise when he takes it. After he reliably takes treats, it will then be time for step up training. But you're not there yet and would likely not be there for weeks-months. No rush though, you're building a lifetime bond with this creature take it nice and slow
Dietarily, if he's eating pellets he's probably not too bad off. I'd definitely cut the nuts and seed down to no more than 20% of his diet. You can and should offer fresh foods daily, but don't expect him to eat them right away. Our amazon reacted to fresh produce as though it was radioactive for a long time. Dietary changes for humans are difficult, it's even harder for a bird who doesn't have the faintest concept of "health". Once you've built up some trust, then you'll be able to work more with getting him to try foods because they'll start following your example once they feel like a member of your "flock".
Good luck and be sure to share some photos of your new friend!