Ever since I unlocked my first relic slot, I always run 'Take More Damage' and I've never taken that relic off. It's usually easy to manage and I don't notice much difference anyways. The relic says "take more damage when injured", so maybe you take more damage only when your health drops.
No Machines is another good one, the only thing I use from that category is the riot shield anyways.
The only relic I never, ever run is 'Do Less Damage', but that's just because of my play style. I use weapon specialist 99% of the time so I'm used to having my guns deal most of the damage and anything that reduces damage feels weird to me.
I've also been using Stand Your Ground a lot since I'm usually running Feral, and Pistols only ever since I bought the pistol arks.
Take More Damage - this relic does not take any affect to body armour and can be easily negated by using it. people claim it affects team mates but letting you know it does not and those players are stupid and noobs.
Do Less Damage - this is balanced out by using weapon specialist as it makes your bullets more powerful but also fire ammo will do damage over time and this is a nice way to get that extra damage while killing.
Pistols Only - I don't recommend this as I would advise you to be a team player and any weapon challenges appear... you cannot participate, if the majority of players use this then there wont be any weapon challenges except pistols but I still don't recommend it.
Half Wallet - I don't recommend this aswell for so early but if you are not running any support or ammo drops then maybe this can easily by pass, if you are the team support then it is hard to earn money to drop them for the team mates
Mortal - this only depends on your skill, people may leave the lobby because it is noticeable though.
Fragile - being a medic and having feral instinct would dramatically increase the health regen and just alike take more damage, run body armour.
Stand your Ground - run feral instinct
No Machines - if your a fast paced killing machine then these aren't as needed
Limited Ammo - I dislike people using this as they constantly need to drop ammo so I don't recommend this unless your using a portable/grenade turret throughout the game but those items I only recommend for score runs as many members including myself aren't fond of users who use them on a regular basis.
Earn your Keep - this is a HELL NO for co-op, during 9 relic runs for the leaderboards we don't ever use this relic too. it is the worst but for solo, it'll be the 9th relic if you make it that far. mayday is too easy anyways
My current go-to loadout is WS, Pistol, Incendiary, Armor
Relics: Take More Damage, Fragile, No Machines, Limited Ammo
IMO personally the 4 easiest relics to run. You will find that PoC becomes a breeze because none of the relics limit your offensive capabilities (assuming you have teammates constantly dropping ammo) while being the Armor guy will ensure you have constant stream of armor, voiding Take More Damage.
Only problem with Take More is you will get the occasional idiot who thinks it still affects the whole team. IW really needs to fix that typo.