What order to do things in is covered in this financial flow chart.
ukpersonal.finance/flowchart/
Moneysaving Expert also has a really good budgeting section
www.moneysavingexpert.com/family/money-help/
While you're in debt, you need to minimise the interest you pay by transferring to low rate credit cards if you can, and paying off as quickly as possible.
But you also need to be realistic about your budget, as you've recognised, you need to save money for annual and irregular expenses like Christmas, insurance, car costs etc etc
However, if you can clear a credit card, you could make that part of your emergency fund because you'd only have to pay for car repairs if they happened, so you'd only pay the credit card high interest rate from that point forward, rather than having the money available in a savings account paying 2% and paying 20% interest on a credit card because you've got a higher balance IYSWIM?
With credit cards, don't just pay the minimum payment, ideally pay as much as you can, but at least £1 more than that, or set up a standing order for whatever the minimum payment is now, and it will pay it off much faster as the balance decreases. If you leave it at the minimum payment, it will take decades to pay off.
With savings, you just want the best instant access account. You won't earn much interest, but it will be some. There are some accounts that pay slightly more as they're marketed as 'rainy day' accounts, so you're only allowed to take money out a few times a year, but that might work quite well for you. You can get the money back whenever you want, but the thought of losing interest if you take money out willy nilly might put you off.
Of course, you can have as many accounts as you like, and they can be very easy to open with online banking, especially if you already have a relationship with the bank, so you could open different accounts for different purposes.
Have a look at:
www.moneysavingexpert.com/savings/savings-accounts-best-interest/
But if you are receiving universal credit, then get a Help to Save account, because it pays a 50% of the highest balance over the 2 year period, so unbeatable if you qualify and well worth having.