They're similar, but not quite the same. 'Go for a walk' suggests a single short excursion, of anywhere between 5 minutes and several hours. It's like 'go for a coffee' or 'go for a pizza' or 'go for a swim' - it tells you the purpose of an individual outing. 'Go walking' puts the emphasis on the activity itself. Like go running, go jogging, go sailing and so on, it is physical activity. Someone could go walking in the mountains for months on end. This is very different from just going 'for a walk'. And, yes, 'go walking' is most certainly grammatically correct!

Yes, same meaning and correct grammar: - I go walking in the park every day. - I go for a walk in the park every day.

Hi, Christine! But this is an expression in an American show Let's Talk in English...

'Go walking' is grammatically wrong. :)