Left and Right Cosets

I find it a lot easier to use cycle notation when working with the symmetric group. example $(1,2,4, 7)\in S_7$ is the permutation that takes 1 to 2, 2 to 4, 4 to 7 and 7 to 1. The unmentioned numbers are left the same.
Your coset $H=\{(1), (1,2)\}$ it has size two and $|S_n|=n! \Rightarrow |S_3|=3!=6$ . So you should expect 3 cosets each of size 2. Basically you just gotta multiply them out and see what happens.
Here are the left cosets of H
$(1)H=\{(1), (1,2)\}$
$(1,3)H=\{(1,3), (1,2,3)\}$
$(2,3)H=\{(2,3), (1,3,2)\}$

Right cosets of H found similarly
$H(1)=\{(1), (1,2)\}$
$H(1,3)=\{(1,3), (1,3,2)\}$
$H(2,3)=\{(2,3), (1,2,3)\}$

Compare these cosets and see that the last two do not match up, so these are not the same. In particular this tells you that H is infact not a normal subgroup of $S_3$ because $(1,3)H \not = H(1,3)$