Yeah, it's kind of a random category. Kohlrabi looks more like a root vegetable, but that bulb is actually an enormous swollen stem (it also tastes a lot like a broccoli stem). But the vegetables included here actually cook fairly similarly, requiring less time than a root vegetable, but a different treatment from a hearty green, although there is certainly some overlap. All these vegetables are versatile too, equally delicious when raw, steamed, sauteed, or roasted with a tender inside and a crispy brown char...ok, perhaps we have a preference.
Most stems, stalks, buds and flowers can be store loosely wrapped in bags in the fridge for at least 3 to 5 days, if not longer. Use your judgment - if the vegetables are a bit wilted or have some tiny brown spots or have lost some vibrancy but still seem fresh and smell like they should, you can probably eat them, just remove any bruised spots with a small knife. Roasting is a good way to cook slightly older veggies, as the crispy edges will help add texture that's been lost due to age. You can also place them in an ice-water bath to revive them a bit, just like salad greens. Once the vegetables start to smell bad or collapse or turn weird colors - time for the compost bin.
Asparagus is a bit fussy - to keep it fresh as long as possible, store upright in the fridge with the bottom ends in a water-filled cup (like how you'd store fresh flowers), then eat quickly. You can store celery this way too.
Most of these vegetables can be frozen, but will lose some structure, so you'll want to eat them cooked after defrosting. To keep them at their best, chop them into small pieces, blanch according to these helpful instructions from The Kitchn, dry and freeze in a sealed container.
Sometimes we get caught up in eating only broccoli florets or fennel bulbs and we forget that there's so much more to all these vegetables. The broccoli stalk is delicious, nutty and sweet - it just needs a good peel of the fibrous outer layer. Older kohlrabi also needs a peel, but then you can eat the leaves just like any other Heartier Green. You can totally eat cauliflower leaves and the thick core, just cut it into thin pieces. Fennel fronds can be used just like dill, and you can save the fennel stalks to flavor your Vegetable Stock.