Firstly - congratulations!!
OK, car hunting.
Decide if you want to purchase from a car dealer or privately.
The pros are:-
Dealer - car guaranteed, usually comes with 1yr warranty, will also be serviced and MOT'd before you get it so you know it is totally roadworthy, however, if something were to go wrong, you are within your legal rights to take it back and get a refund (there is a set time period for this, your Citizens Advice or Trading Standards should help more). Of course, prices will be a smidge higher.
Private - the price will be lower and most sellers will hand the car over there and then to you if you do the test drive and decide to purchase. You can also haggle, most dealers don't really do this. However, if the car goes wrong, you have absolultely no come-back and of course, the car may not have been serviced or MOT'd for almost a year which means more expense for you soon after the car purchase.
Assuming dealer purchase,
test drive.
Make offer.
Pay deposit.
Go home, go to comparison websites, type in the car registration, your info and get prices on insurance.
Go for the one best suited to you (it will be high being you are new to the roads).
You can print off a cover note or the full info will be posted to you in a few days.
Most dealers do a service and MOT in the week before you collect your car, most will also arrange a tax disc for you.
When you go to collect your car, take all your insurance bumph, driving license and remainder of the money.
All done.
If buying privately, it's the same kind of deal but you'll have to get the tax disc yourself.
I'm assuming the car has a bot of the road tax left on it as to get a tax disc you need ther V5 log book which you only get once you've purchased the car.
You can do this on-line but it'll be quicker to go to the Post Office and wait in line.
You need the V5 log book plus your insurance cover note.
MOT is an annual thing, there should be a note somewhere in the owners file - I have a file containing all the service info and previous (and current) MOT's. You get the new one done when the current one is about to expire. To prevent an un-necessary fail, check all lights work - I had a car fail on a dead break light bulb!