But pure guessing is not good too you should use some known properties about (finite) groups:
1)A subgroup must contain the identity element.
2)The subgroup must divide the order of the group.
3)The subgroup of a cyclic group is cyclic.
There might be more well-established facts about subgroups but those three are the most basic ones.
So in the Klein-4-group the possible sizes for a subgroup are: 1,2,4. Because those are divisors of 4. 1 being the trivial group. 4 being, the other extreme, the improper group. While the 2 are all intermediate groups. Now it needs to contain #1 which means there are 3 possible elements left to make a 2 element subset. So there are three possibilities to check. Easy.
With all subgroups must be cyclic. So create all possible cyclic subgroups and those must be a complete list.