Welcome in the multilingual world of cosmetics! As it stands, there are German words for all these things, but sometimes the English (or "Denglish") word is used more often. So let me give you an overview:
Shampoo (sometimes Shampoon or Schampon) has become the standard term, replacing the more oldfashioned Haarwaschmittel.
Here a plethora of terms ares used: (Haar-)Spülung Pflegespülung, Conditioner or, more intense (Haar)-Kur. Marketing speak has come up with a few more fluffy terms like SOS-Repair. For everything that is beyond the apply-and-rinse kind, all bets are off and it's most likely some English phrase or English-German mixup like Leave-in Kur.
Cleaning your body:
There is the good old Seife (soap), that has seen a bit of a revival lately. But if the bar-shaped thingy is not genuine soap it may be called Waschstück or Syndet, both being rather medicinal terms. In liquid form it's either Flüssigseife or sometimes Cremeseife (in the dispenser on the sink) or Duschgel (for the body). And if the professional language maulers (aka advertisers) had a go at it, it may say Body Wash or something completely different on the bottle, leaving you to guess/infer the intended use. I just found two bottles in my bathroom that say "Dusche" and "Erfrischungsdusche" but contain shower gel, not a shower stall. Sigh.
Here it's mostly the texture that decides between Rasierseife, Rasierschaum and Rasiergel.
Surprisingly, this is where a simple Waschmittel will suffice, sometimes specifying texture again, like Waschpulver, Flüssigwaschmittel or Waschmitteltabs (the oversized "tablets") or detailing the use as Vollwaschmittel (for whites) / Feinwaschmittel (delicates) / Colorwaschmittel (Denglish rearing its ugly head again). If you like you can also add Weichspüler to soften your laundry (Caveat: this is not for your hair unlike you assumed in your question!) or Fleckensalz or Bleiche / Bleichmittel to remove tough stains.
Of course you may always make up phrases like "X to wash Y" - there may be the occasional smug salesperson, but mostly you will get the desired results and extra points for creativity!