Assuming you have a six-month Assured Shorthold Tenancy, you don't have to give any notice at all. As long as you are not in occupation after the end of the tenancy (which means moving out before it ends), that's it. For example, this page from Shelter says:
The general rule is that the tenant can leave on the last day of a fixed-term tenancy without giving notice, and this will end the tenancy. If the tenant remains even a day longer than the last day of a fixed-term tenancy, a statutory periodic assured shorthold tenancy will arise, which the tenant can end by serving a valid notice to quit.
 Right d. Flower v. Darby (1786) 1 T.R. 159; Cobb v Stokes (1807) 8 East 358.
This is extremely bad news for landlords, who don't (necessarily) get any notice that they have a void. In general, I would say you ought to give a month's notice (full disclosure, I am a landlord), on the other hand, if they haven't fulfilled their responsibilities perhaps not (but you may have an exaggerated idea of their responsibilities).
If you do this, don't expect to get a good reference from your landlord! (Which is going to make it harder to find somewhere else to live). You should also anticipate difficulties in recovering your deposit (you are legally entitled to it back, but that doesn't mean your landlord can't be difficult about it - possibly even requiring you to sue for it).
I have just noticed the second part of your question.
If you do nothing (in other words, stay in residence), at the end of your Assured Shorthold Tenancy you will switch to a periodic tenancy. The landlord can't force you to switch to another six-month contract, but on the other hand, if you don't, they are perfectly entitled to give you two months notice. If they are smart, they will give you this notice now, so that you can only stay for two months on the periodic tenancy.
Note that the letting agent has an incentive to get you and the landlord to sign another contract (they can charge the landlord a fee for it), so they may not be being entirely honest about whether a periodic tenancy is an option. (On the other hand, a poor landlord may be more interested in locking a tenant in for six months.)