'I'm going shopping' should also work, though currently not accepted, if for no other reason, because one of it's possible meanings / uses is to indicate that you will do the action in the future. What are doing tomorrow? I'm going shopping.
Pretty much, yes. Slight difference in nuance may be that 買い物 (かいもの)をしに行きます puts more emphasis on the activity (e.g. because it's something you have to do, or really enjoy doing), while 買い物に行きます is the standard expression.
Your remark about 買い物 not being the direct object is quite right, but 買い物 means 'things purchased,' as well as 'shopping' and is therefore broader than 'groceries,' which could be translated as 食料品 (しょくりょうひん).