I've had to do this in the past 3 years, for both mum and dad, who died only 7 months apart. You have my condolences.
+1 for getting multiple copies of the death certificate - 10 is a good number. You'll be surprised at just who needs them. A few copies of the will will be a good idea too - but IIRC that needs to be certified by a solicitor - banks etc. won't accept a photocopy.
Beyond the immediacies of the funeral, some actions will depend upon if there is a surviving spouse, as (depending on the terms of the will) the estate (or most of it) usually passes to them, without any tax issues. If your dad was the last spouse, then you may have tax issues to deal with if the total value of the estate was >£650k. It's important to find out what the assets are asap: Building society accounts, bank accounts, shares, property etc. You may need to do a bit of digging through papers.
You need to tell any pension providers about the death so that pension payments can be stopped. Tell the local authority if there's a now empty house as the Community Charge may be reduced.
If you had access to you father's affairs during his lifetime via an Enduring Power of Attorney then the mechanics are much easier as you will have had access to bank accounts etc. and know what assets are where, although that access ceases on the death of the surviving spouse. Once you tell the bank etc. that the account holder has died (unless it's a joint account and another signatory is still alive) they'll freeze the account and even if you have an EPA, you can't get at the assets until you have Probate.
Gaining probate for the estate is done by the executors - there were 4 for my parents' estate but I (as one of them) did it for us all. The forms are downloadable from the Probate Office and it's all done by post. Once you have the details of the assets, and have filled in the forms it does not take long (less than a month for me). If Inheritance Tax is due (40% on assets >£650k) (roughly), then that has to be paid before they'll give you Probate (so someone may need to stump up cash to pay the IHT before you can get access to the remaining assets). If you think IHT may be due, then get the forms from the HMRC website - there were 90 pages of these for my parents' estate - they are not complicated but there are a lot of them. You need to prioritise sorting the IHT.
You may need to set up an executor's bank account to receive the funds from the estate once you have probate - we did this with the bank my dad had has his accounts with. They'll give you a cheque book etc.
Liquidating and distributing the estate may not be quick or simple, there are a number of hurdles to be jumped through, but with patience and attention to detail, you'll get through it and come out the other side. It took me about 9 months from the second death to distributing the monies from the estate to the beneficiaries, and then close the bank account.
Whilst it was a difficult few months, and sometimes it felt as though we'd never get through it, everybody I dealt with was helpful and although they needed bits of paper before anything could be done, it was all for a good reason and eventually you get through the process.
Being an executor is a fairly onerous task if you are the one who elects to do the leg work. However, I saw it as the last thing that I (as the eldest child) could do for my mum and dad, and I'm pleased I did.