Going against what was just said, but there's no shame in buying frozen vegetables, especially peas and corn. They're quite nutritious and easy to store.

Fresh vegetables can stay good too if you buy the right kinds. I've found that asian leafy green vegetables like yu choy or gai lan keep a lot better. The leaves may turn a little yellow but they don't end up a slimy mess in a day. Cabbage is the western equivalent that keeps forever.