Most carrots need a lot of growing space or very deep pots, you need the short, stubby varieties. TBH, you can grow most vegetables in pots, you just need to pick the right variety that is suitable for container growing. This Kew Garden range is specifically for container growing:
www.thompson-morgan.com/brands-kew-urban-gardens
I would look into companion plants that deter pests or add nutrients to the soil eg tomatoes and basil, mint or marigolds, nasturtiums (also edible, like rocket) to lure away blackfly etc. It also looks prettier so your pots are attractive as well as practical.
www.gardenersworld.com/plants/companion-planting-combinations/
Your vegetables do need nutrients though, so many will need feeding for the best crop if you are using old compost. Nasturtiums will thrive on poor soil, they will flower more instead of growing leaves. Salads, beans, peas, root vegetables, herbs, squash and courgettes should be fine. Your tomatoes will need new compost though, as will aubergines, peppers and chillis.
If you don't have much space, cut and come again salads and herbs are good because you don't have to sow another crop after harvesting. Ditto quick to harvest veg like radishes that are ready in a few weeks. Don't forget to plant seedlings ready to grow on in the pot after your first sowing has been harvested so you get successive crops.
Supermarket herbs can be split up and grown on, and are easier to grow than from seed.
www.gardenersworld.com/how-to/grow-plants/how-to-divide-supermarket-herbs/
You could also consider fruit like strawberries, raspberries, blackberries and blueberries.