The link has good detailed description on 2-sylow subgroups:

http://sps.nus.edu.sg/~kokyiksi/roddyurops.pdf

However, for S6 we can directly get the 2-sylow subgroup.

Consider, S4. The 2-sylow subgroup in this group has order 2^3 = 8 and is given as:

P = {(1)(2)(3)(4), (1 2), (3 4), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 4 2 3), (1 3 2 4)}.

Since S6 has 6 characters {1, 2, 3, 4, 5, 6}. Consider:

Q = AP. Where A = ((5 6)).

Clearly, Q is of order 2^4 = 16 and is the required 2-sylow subgroup.