Official Errata and FAQ 1.0: https://whc-cdn.games-workshop.com/wp-content/uploads/2017/09/Codex_Adeptus_Mechanicus_ENG.pdf
Canticles of the Omnissiah
At the start of each battle round you either roll on the Canticle table or pick one you haven't picked before. Any extra rules the activated Canticle gives your units will be in effect immediately. All but one of these are pretty straightforward, more on that one later.
Incantation of the Iron Soul: Your army gains ATSKNF. It's a fine rule to have, but it only helps with full sized Skitarii, Sicarian and Electro-Priest units. Those units gain more from other Canticles, so this one is situational at best.
Litany of the Electromancer: This is the weird one. Does it only work against enemy units in close combat at the start of the battle round or does it work whenever enemy unit comes within 1" of your units? The first FAQ doesn't address this issue. Or maybe I'm the only one who doesn't understand this? Anyway, roll a D6 for each enemy unit in close combat and on a result of 6, deal D3 mortal wounds to that unit. Not great, unless you manage to lock your opponent's entire army into close combat at the beginning of a battle round and then get few lucky rolls.
Chant of the Remorseless Fist: Reroll hit rolls of one in the Fight phase. Good for melee army.
Shroudpsalm: This is what you almost always want; everything counts as being in cover. You'll get the +1 to armor save even with vehicles! No need to have them at least 50% obscured by something. This is the Canticle to pick on first battle round, usually.
Invocation of the Machine Might: More strength for your units. Good for melee army.
Benediction of the Omnissiah: Reroll hit rolls of one in the Shooting phase. Great backup if your Dominus dies or gets lost somewhere.
Forge World Dogmas
Mars: This one is good if you want to fish for that elusive Shroudpsalm every turn, or if you want to include Cawl in your army. Both of those work well with the other...
Graia: FNP, but different. Use this if you play lots of Skitarii and Electro-Priest units. With this rule, Electro-Priests have 5+/5+/6+ saves against everything!
Metalica: Advance and shoot assault weapons (rapid fire becomes assault, too) without penalty. Skitarii horde, anyone?
Lucius: More durability for your units against Bolt Rifles, Heavy Bolters, Autocannons and such. It's great if you mostly play against Marine armies.
Agripinaa: Doubles the effectiveness of your overwatch. Probably the worst of the Dogmas.
Stygies VIII: This is the Dogma for power gamers. It makes your whole army more durable in firefight.
Ryza: If you want melee centric AdMech army, this is great. I painted my army with Ryza color scheme during seventh edition, so I'm sticking with this Dogma.
Next time I'll take a look into Warlord Traits and HQ units.