I did a few things when I started working.
First, I'm very conservative and got hired on during the 2008/2009 financial melt down. I saved up enough cash to survive for a year (I was still used to college spending so this didn't actually take long to accumulate). I put this cash into 4 CDs, each is a one year CD and one comes up every 3 months. The interest isn't great, it's pretty bad, but it's better than the interest on a savings account. This isn't an investment, this is "in case I lose my job or get furloughed or ...". I've since made it a point to bump up the amount in the CDs to represent my current cost of living less things I can easily cut every couple of years.
Second, I owned my car outright at the time, I put the equivalent of a car payment (it was old then) into a savings account every month. I used this to (almost) pay cash for my new car this past year.
Third, I put a similar amount into a brokerage account, but I didn't start that until after I had that 1-year fund. I'm not a great investor, I can't give advice on this part, I've just gotten lucky on a couple things but mostly played it safe.
Fourth, I know how much cash I need in my checking account every month (with a buffer, just in case), so at the end of the month I transfer the excess to my savings. (This year excepted, a bunch of unexpected medical expenses)
The result is, if I lose my job, I have enough cash on hand to get me to my first CD's maturity date, and they can each carry me to the next. This is a lot of peace of mind for me after seeing many colleagues laid-off in 2008/2009 and others who'd been unemployed for a year or more. In my current job that's a low risk, but once the money is saved you can ignore it (until you need it). I also have enough cash + credit that, in a catastrophe, I could carry myself to the CD maturity dates to pay off the debt or, worst case, I could take a hit on the CDs and close them early (I wouldn't lose much, penalty is the interest + a fee and the interest is low at the moment).
This next part is US-centric, I don't know the UK's retirement system.
Retirement, here in the US we have (at many companies) 401k's available with some matching. I put in the minimum needed to get that matching: it's free money for retirement. I put in more now that I'm in a better paying job and have my other finances in order. We also have IRAs, another retirement account, I've been maxing that out since my 3rd year of full-time employment, I could have started my 2nd year but decided the 1-year fund had priority at the time (I don't regret that decision, my employer at the time laid a lot of people off that year and my office was next on the chopping block). Whatever the UK equivalent is, explore them. I know too many of my peers that have no savings or minimal savings for retirement, we're obviously 20-30 years away from it, but investing earlier you'll get a much greater return, and when you have a family and need money for those sorts of expenses, you can (though you should try to avoid this) cut back on your retirement savings without too severely impacting your retirement goal.