If you're talking about groups with + not *, I think I can help...
The cardinal of the ideal divides the cardinal of the original set, here it's Z/6Z
Hence, the ideals have possible cardinals : 1,2,3,6.
This also helps in finding the elements of the ideals : starting from 0, the following element of the ideal will be , if we work in Z/nZ.
For example, the ideal of Z/6Z which has 3 elements will be the set of elements x such as :
This means that the ideal of Z/6Z with 3 elements is :
I hope this is clear enough