Hi jclo
If you don't know whether you like asparagus or not, your inherited plants provide a good opportunity to try them. Next spring, that is.
At this moment you just leave the plants as they are. Cut down the fern to ground level when it starts to wither and add a layer of compost so the plants are covered for the winter.
Do not put the ferns in the compost bin. Asparagus can carry pests and diseases that are contagious, so my allotment neighbour tells me.
For next spring: you can either go for white asparagus or for green ones.
For white asparagus you need to cover every tip that shows above ground. Cut them when you think they are about 25 cm. long.
If you don't go to your aloltment everyday, green asparagus are much easier: just let the tips grow. Cut them before the top starts to open, otherwise the asparagus will be woody.
The berries contain the seeds, sow a few in the asparagus bed. If you like the veg, you want to have young plants for the future. And if you discover you don't like asparagus: just pull them out.
Good luck.