Whenever I have gone I've usually got there around 10am and have never had a problem with parking. I can't say which days are generally quieter.
All you need for a readers ticket is 2 forms of id, a driving license and bank card would suffice. There are lockers downstairs for all your bags and coats etc which are free to use. You will find some clear plastic bags to take anything you need for research purposes, such as pencils, notepads, cameras and laptops. You will not be allowed to take pens into the reading rooms. You can take photocopies of microfiche or film, to do this I think you need to put credit on your readers card, but I'm sure someone will correct me if this is wrong. There are plenty of staff to give you help and don't be afraid to ask, it is quite intimidating when you go there for the first time.
Once you have found anything that is a paper record then you have to order it using your readers ticket and then go to the reading rooms to view it. You have to hand over your bag to the security men and scan your ticket to enter the reading rooms. Once inside you either collect your record from a pre-chosen number (done during the ordering process) which corresponds to a locker and seat which is yours for the duration of your stay. In the reading room upstairs you just have to go to the counter and quote your number. This is generally for oversize records. You are allowed to use a camera to make your own copies if you want, but they do offer a photocopying service if required and except card payments.
It is actually a great day out (well I think so

and it's amazing to think of all the history contained within the building, so enjoy your visit.