You can do Patrol Mission. If you have an XP boost you can level fairly quickly using that and doing Patrols.
STO is about my Liberated Borg Federation Captain with his Breen 1st Officer, Jem'Hadar Tactical Officer, Liberated Borg Engineering Officer, Android Ops Officer, Photonic Science Officer, Gorn Science Officer, and Reman Medical Officer jumping into their Jem'Hadar Carrier and flying off to do missions for the new Romulan Empire. But for some players allowing a T5 Connie to be used breaks the canon in the game.
You can also do the Mirror Incursion when it becomes the active event. It usually appears every 6-9 hours, and you can usually get 3-4 levels out of the hour-long event as well as about 4000 dilithium with a good group.
I have a suggestion on leveling a character and that would be to do the story missions. I have never got a level up in mirror event and most of the time I miss it anyway.
I know the story missions suck and they seriously need to revamp it but that wont happen since every update is more toward end game content and new rep systems that make us enlessly grind until we want to quit.
I recently leveled a new character to max and if you work hard on it then it isn't that bad.
As for the rep system and all of that to make the character the best you can have I do have a tip on that as well. I used my main toon to get the gear when my alt was leveling up. This gear was not from the rep or fleet because that is bound to character.
Using the account bank I was able to transfer gear to the other character. It really made a difference leveling up and once I hit level 50 I slapped some mk xii gear on the character. I then ran a few Elite STF's to get T3 omega and the ground maco set.
As for the rest of the leveling up with the doff system and all of that, I am working on it but the main part of levelign up is done.
Hope this helps.
One thing to be aware of in the Mirror event is this, if you join an event in which a higher level Tier toon is in there (helping a lower level friend, which it does not do in Mirror btw) immediately leave. The presence of a higher tier character will drastically reduce the experience earned from each kill.