We don't normally say 'do shopping'. In British English, you can say 'do the shopping' to mean buying household goods and groceries. This phrase isn't used in American English. You could also say 'do some shopping', but not simply 'do shopping'. 'Go shopping' suggests a more enjoyable leisure activity, including window-shopping, buying clothes and souvenirs and so on. As for 'party', no, we don't say 'make a party'. If you were to write that in a test it would be marked incorrect. The most usual phrase is to 'have a party'. For example, 'I'm having a party for my birthday. Can you come?'

You go shopping and have a party.You can "do the shopping" which means buy things that you need for the house,like food.But "to go shopping" is more like a fun activity when you just go with friends and buy clothes,or even don't buy anything but just look and try things on.At least that's how I use it (in the UK)