You won’t save money - it’s often quite an expensive series of failures - but it’s fun and the veg tastes great.
Money-saving things to grow are cut and come again salads, herbs like coriander that grow extremely easily from seeds, spring onions and radishes and rocket (because they are super quick to give you a yield and you can get 5 months of salads from a couple of packets of seeds).
Tomatoes are much harder - they germinate well from seed but getting them to ripen and produce a properly decent crop can involve a lot of watering, feeding, pinching out and keeping everything crossed you don’t get hit with blight the week they start to ripen. So, great idea for a few tomatoes for the fun of eating them warm off the plant, not so great as a money saver.
Long term crops like broccoli, cauli, cabbage, sprouts, leeks, kale etc etc are not a great choice if you have very limited space - it takes many months from sowing seeds to having something to eat. Things like carrots are so cheap to buy and hard to get to a decent size (they split into ‘legs’ if the soil isn’t fine and unlumpy).
Spuds are great and you can grow them in a bag rather than needing to use raised bed space.
Rhubarb, asparagus and other perennial crops are wonderful in the long term but you get nothing for the first couple of years while they establish.
Join is on the Veg Patch threads. Herbs, salads and do on are a nice easy entry into growing your own.