I'm not sure about every pod brand, but the lavazza ones we buy use recyclable plastics, so we just clean them out, but the grounds to use in the compost and recycle the rest. It's time consuming opening each pod, but for us it's worth the luxury of having a pod machine until we can afford a proper coffee machine. DH is a mad recycler as well, so if there's a way, we find it lol.
Ahh so those pods are the type that you can put in the normal plastics recycling bin? - would I be able to tell on a used pod if it is a recyclable one or not (e.g. does it have the symbol or anything). I think I prefer the idea of biodegradable pods, but if you can recycle the plastic ones at home that would be my second choice.

ETA. I have just read some stuff that suggests that all plastic pods are recyclable - as long as separate grounds first.