One thing the new UI seems to be missing is the total +% to various stats. It'd be really easy to add +100% next to Shield-Related: or something similar, because a 10% bonus when you have no bonuses is very different from a 10% bonus when you already have a 200% bonus, since they stack additively. And when you can have a couple dozen sources or so for stats, it's really hard to add up yourself.
Personally, I just look at the new gear after each mission, add it if it is good, ignore it if not. Occasionally I'll find myself using an old piece of gear, but it is generally for one of two reasons: 1. I'm having to mess with power requirements for my exos or 2. I just unequipped it from an exo and another is getting it as a hand-me-down. Personally I don't find all the parts to be so overwhelming, but I'm only playing on hard right now, and I have a good head for numbers in the first place.
That aside, I'll echo other people's advice. You'll generally want to just focus on a couple of stats for each exo. My assault is loaded up with bonuses to regen, damage reduction, and % all damage. I'm well ahead of the damage curve of everything except a few 'one shot kill' type bots like the blunderbot with reduction/regen, so I just let most things run out of ammo while I whittle them down with weak weapons. But, I do the +% damage so that if I do come across something strong, I can potentially destroy it before it gets a shot off, generally with a bit of stealth kiting involved. My siege gets all my leftover +% damage and focuses on AoE/ammo for the room clearing rockets. Ninja is a trapmaster with sentries and a bit of damage and stealth to take stuff out with his cutting laser. And finally my science bot is a virus master with a fair bit of stealth. Just walk up to troublesome bots under stealth, virus them, and let them take care of the level for me.
Of course, that's just me and my team. You could do a sniper and set him up as an alpha striker, focusing on damage damage damage and more damage to destroy stuff before it can fire on you, but that can backfire (as can anything I suppose), just remember to get enough ammo to get through the level.