No, you don't need to open a company. But if you wish to issue receipts, you by all means can. If you're trying to establish yourselves as a 'business' in the public eye, just come up with a catchy (yet professional) name, create your policies and procedures and your fee structure. Perhaps, regarding the taxi rides, you could incorporate that into your fee structure?

Good luck ladies....I'm fairly impressed with your approach to babysitting