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.