How can I prove the sum of squares and the sum of cubes with binomial coefficients? My main problem is starting. I can't "see" anything that might give me an idea to find a relationship between these two things

Thank you :)
If I may hazard a guess, perhaps you are expected to use the identities
$$\sum_{k=0}^n\binom k2=\binom{n+1}3$$
and
$$\sum_{k=0}^n\binom k3=\binom{n+1}4$$
to derive formulas for
$$\sum_{k=0}^nk^2$$
and
$$\sum_{k=0}^nk^3.$$
For instance,
$$\binom{n+1}3=\sum_{k=0}^n\binom k2=\sum_{k=0}^n\frac{k^2-k}2=\frac12\sum_{k=0}^nk^2-\frac12\sum_{k=0}^nk$$
so
$$\sum_{k=0}^nk^2=2\binom{n+1}3+\sum_{k=0}^nk=2\binom{n+1}3+\binom{n+1}2.$$
To prove $\sum_{k=1}^n k^3 = \binom{n+1}{2}^2$, use Pascal's law to find

$$\binom{n+1}{2}^2 = \left( \binom{n}{2} + \binom{n}{1} \right)^2 = \binom{n}{2}^2 + n^3  $$

and use induction. You're likely to find the formula for sums of squares in a similar fashion.