With Exigency you want to do some research on each ship you are going to use/pilot, because each ship has a separate right-click ship system that you need to go in the codex to find out about. (iirc) Maybe someone more well versed with the content, like Meso, could do a quick overview post of the various right-click abilities.
The ship system slot is taken up by the Reflector, a cooldown limited, near-perfect 360 degree defense that also reflects incoming fire and takes over incoming missiles. Also, practically every Exi ship has built-in micromissile spam that puts a timer on your opponent to finish the fight fast, because once armor is gone they will shred everything.
Enemy faction info:
Shadowyards - can be hard to fight fleets with lots of Morningstars early on, they are like mini Apogees that will kite you and make your life miserable, and their hard shields present a problem. Also look out for carrier-heavy fleets or cruiser-heavy fleets. Overall, similar to fighting Tri-Tachyon.
SCY - each fight will probably surprise you, some ships are very hard to kill 1v1 because they can run away from you while their front shields soak damage, and then they'll vent, giving you a small window to hurt them. SCY ships basically only get rid of flux by venting, but they have a ton of capacity. Try to get a surround. Overall, ships are fast and fairly specialized.
Blackrock - try not to fight if outnumbered, each ship has a lot of firepower and mobility, so the ships that aren't taking damage will flank and tear through undefended targets. They have bad shields and will vent when pressured so try to work in some damage. Also exi and blackrock counter each other in weird ways that can make it a stomp either way. Look out for PDE weapons when you are in Exi ships.
Templars - Wear diapers