You will be responsible for up to the remaining rent as long as the landlord tries to rent it out and can't. The landlord is responsible for trying to rent it out to reduce these costs as much as possible.
They will charge you for advertising costs as well, such as putting the ad in the paper.
If your landlord is a nice person and you don't anticipate a problem, just let them know you will move out, you should give as much notice as possible. I think you will be responsible for the 30 days if you don't give a proper notice as if you were moving out at the end of a lease even if it gets rented right back out. If you have a good reason for doing so, one that the landlord will understand to be good, then they will most likely be helpful. (lost job, move out of state to help parents, things like this that are unavoidable, can't afford it might be something they would understand too but will be less sympathetic.)
So you will owe the last month, plus advertising costs, plus of course any thing from the deposit needed to clean it up or fix it for the next tenant, PLUS the rent until it is rented out or the lease expipres, UNLESS the landlord does not make an effort to rent it out. Usually this will mean they have to at least try as hard as they did to get you in there.
Good luck.
here is a snippet of the connecticut law that applies:
Sec. 47a-11a. Abandonment of unit by tenant.
(a) If the tenant abandons the dwelling unit, the landlord shall make reasonable efforts to rent it at a fair rental in mitigation of damages.
(b) If the landlord fails to use reasonable efforts to rent the dwelling unit at a fair rental, the rental agreement is deemed to be terminated by the landlord as of the date the landlord has notice of the abandonment.