The British in Poland blog pretty much answers all your questions in detail, however...
I would really appreciate someone could offer guidance on how I must apply at first for temporary residency. I am aware of the office I must obtain and fill out the forms but what documentation do they require. For example what type of documentation confirms my stay in Poland and how many pieces of this documentation do I need?
For the temporary (three month registration), you just have to go to the Urząd Miasta with the owner(s) of the property that you wish to be registered at and fill in a form with your passport. It's painless, as long as you've got the owners of the property with you. You might have problems with getting the owners of the flat to agree however - which is why it's imperative that you get them to agree to this before handing over any sort of rent or deposit.
For the five year residence permit, it's more hassle - but you don't have to worry about this right now.
Secondly how do I obtain a Polish national insurance number?
There's no such thing as a single number - you have to obtain a NIP number (tax number) and a PESEL. The NIP is a doddle to get - once you're registered, go to the tax office (Urząd Skarbowy) and fill in a form and wait three weeks.
The PESEL will come after you obtain the EU residence permit for 5 years - but it isn't needed for day to day life. It's the closest to our national insurance numbers, though. If you do need it, they'll often just accept your passport number as a substitute.
Finally how easy is it for a UK citizen to open a Polish bank account? Do you recommend any banks? Are any of these accounts open to holding foreign currency (GBP) although this is less of an issue?
mBank all the way. They offer multiple accounts (in PLN, USD, GBP, EUR and CHF) for free, along with almost free banking. They don't need proof of residency, just a declaration that you're actually resident in Poland - and they're very pain free to setup. They've also always got an English speaker on the phone (in my experience!).