BINARY DOMAIN

For anyone struggling with this, there actually is an infinite you can exploit to max Cain's trust (although it is a little harder than the others). During chapter 4-4, when the Resistance's robots get sabotaged, you will have to take down a large robot (yellow, has 4 arms). During this fight, generic green robots will spawn infinitely, in groups of 3 or 4. To minimize the damage the giant robot does to you and your team, take 3 of its arms out, so it only has 1 left. Your teammates are unlikely to kill it on their own, and you can shoot the generic robots for easy trust gain.
The extra trust gain nanotech is useful in this, and "Cakewalk" difficulty is also recommended.
EDIT: as pointed out in comments, this isn't an easy infinite on harder difficulties. Cakewalk is almost a requirement.

You want to have Cain in your party from the moment he becomes available until he leaves. Make sure to pick up the trust booster nanos from the guy in Shibuya in chapter 2-3. When you have to choose between him and Faye, and you also trying to max out Faye, choose Cain over her. If you've been using Faye since the start, she'll be ahead of Cain in terms of trust.

You will have a few dialog choices where you can boost his trust in you. Just agree with him. However, this is in not sufficient to get his trust maxed. You'll need to impress him and your comrades in combat.

The easiest way to boost your group's trust is with chain kills, headshots, and one-shot kills. The RAOW-S7 and Yamato-51 shotguns make it very easy to get all three. Whenever you can get the shotgun, grab it and keep it as long as you can. You should only ever switch it out for the sniper rifle or RPG. Once you're done with it, go and swap back to the shotgun.

When you come across a group of enemies at close-to-medium range, switch to it and charge up to them when you have an opening. Aim just below the head, and fire at point-blank. This is usually enough to get both a headshot and one-shot kill on anything other than the Golems (black shotgun-users) and security shield-using Jarheads. Your team will quickly gush over your rapid kill streak. Make sure you have medkits with you when doing this, as you will get downed a few times. You don't want to rely on them to rush out into a firefight to help you.

Other trust-related events:
When you are told to follow the group to the truck in 4-4, go immediately. Staying behind will steadily decay your trust. It's not worth it, tempting as the credit payout would be.

In chapter 4-5, if you can finish the howitzer battle with more than 25% integrity, you'll get a trust boost for your team. Make sure to keep spinning around looking for Hercs. Take them out first, then scan for other targets. Keep shooting the Grand Lancer in the head over and over, taking time to aim each shot.

In chapter 5-1, take out the first group of Shinobis as fast as possible. They can flash step around your bullets, but they cannot dodge your Shock Burst. Down them with the Shock Burst, then lay into their legs with your assault rifle or the shotgun. If you take out all three fast enough (before Shindo makes his suggestion), you'll get another trust boost for everyone.

Going crazy with the shotgun got my trust with Cain maxed in the middle of chapter 5-2 on No Mercy difficulty. Faye's trust got maxed just before then.

Keep in mind you will likely not be able to get everyone's trust maxed in one playthrough. I got Big Bo and Rachel maxed in the first on Survivor, and Faye, Cain, Shindo, and Charlie (in that order) in the 2nd on No Mercy.