what's the best weapons and shield against romulan mogai ships
There's no really one type of weapons and shields that are super effective against Romulan Megai escorts. If you are having trouble, I would suggest using a singular type (phasers only or disruptors only etc). That way your skill points can be divided out into other skills that will improve your survival or damage output. As for your shields, I tend to find covariant types are my favorite overall shields. I can't give you more specific advice without looking at your build.
Buff up, hit them hard and fast is my best advice. If you have subnucleonic beam (Sci captains), I suggest unleashing that AFTER the Mogai activates Tac Team and Cannon Rapid Fire or Torpedo High yeild(but hasn't fired it yet). If you have tractor beams, well turnabout is fair play!
Depends on what type of captain you are and what ship you are piloting.
In every case I recommend going into a heavy defensive stance when they hit attack pattern apha. This in-tells running most of your power through your shields and engines and activating defensive buffs. Once the attack patter alpha is over you immediately start an offensive strike.
If you're a good escort pilot, you want to skip the defensive maneuvering, keep your engines hot, put on the pressure and stay out of the frontal firing arch.
Being a romulan ship, you will also want to keep TSS or EPtS in your back pocket so you don't get destroyed by a HY Plasma Torp.