Looking on the internet deeply has found these results:

i went out to shop is the most popular phrase on the web.

i went out for shopping 1283 results on the web	i went out to shop 1466,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
(Well, if I went out for shopping I probably could but I do most of my shopping online, more limited options.) I have never liked stockings, even they make my legs�... Oct 18, 2014 ... Last weekend I went out for shopping, I could see huge rush at the market place as was expected as everyone must have had the same plan�... Jan 23, 2015 ... Lucky I got 2 banana shape from hospital at that time. the pillow also help when I put it underarm while I went out for shopping (I went out a lot�... Apr 19, 2014 ... I went out for shopping in the busy streets and was overawed by the warmth and recognition from the crowds.�. Further elaborating on the�...	It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

i went out for shopping

1283 results on the web

i went out to shop

1466,000 results on the web

More popular!

(Well, if I went out for shopping I probably could but I do most of my shopping online, more limited options.) I have never liked stockings, even they make my legs�...
Oct 18, 2014 ... Last weekend I went out for shopping, I could see huge rush at the market place as was expected as everyone must have had the same plan�...
Jan 23, 2015 ... Lucky I got 2 banana shape from hospital at that time. the pillow also help when I put it underarm while I went out for shopping (I went out a lot�...
Apr 19, 2014 ... I went out for shopping in the busy streets and was overawed by the warmth and recognition from the crowds.�. Further elaborating on the�...

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.