I'm a British English speaker and I'd translate 'faire des courses' as 'go shopping', 'do some shopping' or maybe 'get some shopping'. This could be either in a supermarket or in several smaller shops. An American English speaker would be more likely to specify what type of shopping it is by using the term 'groceries'. Personally, I'd never say 'run errands'. It sounds quite old-fashioned to my ear, but I believe it's used more in the US. It suggests a short trip out from home to buy a few essentials ( not necessarily groceries). The phrase would also cover other necessary trips, for example, to go to the bank or post office, drop off or pick up clothes at the dry cleaner's, and so on.

Mr Google says "faire les courses" --> "go shopping" "to do shopping" == ? to go shopping - I guess as above, to leave somewhere and go somewhere and buy things "To do some shopping" - go spend some time buying things window shopping = go, mostly look, not so much buying. . Sometimes g"go shopping" can include the looking without buying. Context and person-dependent. "to run errands" ==> "errands" are things that need to be done, involving going places, like returning a library book, dropping off that pie to MRs. Smith, Generally small pick up or drop off tasks. Not usuually used for any substantial piece of work. Hence not wash the car (no travel involved), nor going to work (noit collecting or delivering) "to go grocery shopping" = to go shopping for groceries, specifically daily food supplies. to go toy shopping to go bicycle shopping to go shopping for clothes to go shoew shopping = to go shopping for shoes What do you say if "I want to go to the supermarket ? ==> "I want to go to the supermarket" I need some cereal. I'm off to the supermarket. I'm off to Carefour. I want to go to the supermarket on the way home. ... Enjoy.