Having only recently got a car myself I'd say go for it - it's changed my life in terms of how much easier I find things (popping to the shops last minute, not being at the whims of public transport, being able to get to friends/family quickly in times of need, my own space in the morning to ease into the day etc etc etc).
But - if it's to save money, I think you'd be surprised how much a car can cost.
For example: my car is £100 a month (a small car, hire purchase), then my insurance is £80 (newish driver and expensive insurance area, so yours would probably be cheaper!), then for the amount of distance you're covering you could probably get away with 1 or 1.5 tanks of petrol a month, which in my car as an example would be £40-60. Then there's tax (£30 a year on my car), MOT (£35 was my last one, but if anything needs doing that will add up). If you're planning to keep it in good nick, it'll need servicing maybe once a year (or two years, with the distance you'll be doing), which can range but more than £100 a time. This year my car has needed 4 new tyres (£200), and new brake pads and discs (another £200).
These are all standard costs - my car hasn't had any major issues as it's only 4 years old.
There's also the issue that once you HAVE a car, you'll likely drive more than you expect - suddenly there's day trips because it's easier and you can, or you'll find yourself driving places you would have walked or got the bus before etc etc. So you add more in petrol (and miles, so the need for new tyres, servicing etc) becomes more often.
Then I spend £4-5 every 6 weeks or so getting it washed (obviously I could do this myself, but in winter I don't want to spend time outside my house in the cold sloshing water everywhere, and it gets covered in shite from salt on the roads etc).
Then there's parking - can you park at work for free?
I love my car, I genuinely say it's the best thing I ever bought, and I don't regret a single penny spent on it - having my own car has in a roundabout way changed my life (was able to move to new city but keep job), but it certainly isn't cheaper than public transport.