Question

have a great time是doing sth还是to do sth

Answer 1

用doing sth

have a great time doing = enjoy doing
享受做某事；喜欢做某事；做某事很开心；
eg.
We have a great time preparing our own dinner.
我们很享受准备晚餐。
I have a great time playing basketball with you.
和你一起打篮球我很开心。

同理，“have a good time ”“have a difficult time”“have a hard time”等形式的词组后都跟doing sth。
例如：
I have a difficult time mixing work and pleasure.
我很难同时兼顾工作和娱乐。
As usual, the police will have a difficult time.
警察的日子照例不会好过的。
I had a difficult time trying a feather.
我很费劲地摇了一下桨。

Answer 2

后接DOING STH

We always have a great time talking to each other. 我们互相交谈时总是很愉快。
We'll have a great time camping out, I'm sure.
我确定我们会在露营时玩得很开心。