American English says 'back and forth'. British English says 'to and fro'. These aren't especially old-fashioned expressions. They're just standard English, and they're often used in all sorts of everyday contexts. What gave you the idea that 'to and fro' wasn't 'contemporary'?

PS If you really think that ' going to and fro' sounds like something from Shakespeare (which it doesn't) you could say 'back and forth', which is what US English prefers, or even 'backwards and forwards'. 'I was going backwards and forwards to the warehouse all day long.'

Some context would help, but I would usually say "back and forth."