Problem 1. Here's an example: take $A=\mathbb{N}$, $B=\{0,1,2,3\}$, and let $f\colon A\to B$ map the natural numbers to their remainder when divided by $4$ (so $f(3) = 3$, $f(15) = 3$, $f(21) = 1$, etc). Let $R$ be the equivalence relation on $B$ given by $aRb$ if and only if $a^2\equiv b^2 \pmod{4}$.
The relation $S$ is defined as follows: first map to $B$, then check $R$ for the images. If the images are related, then the original elements are related; if the images are not related, then the originals are not related. For example, to see whether $3R10$ holds, we take $f(3)=3$ and $f(10)=2$, and check whether $3R2$ holds or not; since $3^2 \not\equiv 2^2\pmod{4}$, then $3\not R2$; that is, $f(3)\not Rf(2)$, so $3\not S 2$. On the other hand, $7S9$ holds: because $f(7) = 3$, $f(9) = 1$, and $f(7)^2 = 9 \equiv 1 = f(9)^2\pmod{4}$, so $f(7)Rf(9)$ is true.
That's the idea.
Now, for a general case. First convince yourself that $S$ is in fact an equivalence relation on $A$.
Once you do that, in order to check (a), you want to see if $f([a]_S)\subseteq [f(a)]_R$. Well, take an element in $f([a]_S)$; this is a $b\in B$ such that $b=f(x)$ for some $x\in[a]_S$. that is, $xSa$ is true, and $f(x)=b$. So you want to see whether $b\in[f(a)]_R$; that is, whether $bRf(a)$. So, from assuming $xSa$ holds, you want to see whether you can always conclude that $f(x)Sf(a)$.
For (b), you proceed similarly; now take $b\in [f(a)]_R$, and you want to see whether $b\in f([a]_S)$. That is, assume that $bRf(a)$; must there exist an $x\in A$ such that $xSa$ and $f(x)=b$? If so, prove it; if not, give a specific counterexample with specific $A$, $B$, $f$, $R$, and $S$, and an exlicit $a$ and $b$.
Problem 2. (a) If your sets are finite, the your guesses are right; if your sets are not necessarily finite, you're in trouble. Consider $A=B=\mathbb{N}$ and $f(n) = 2n$.
For (b); you know there is a $1-1$ function from $A$ to $B$; see if you can construct a $1-1$ function from $A\times C$ to $B\times C$ (and see if you can spot why you need $C\neq\emptyset$). This will show that $|A|\leq|B|$ implies $|A\times C|\leq|B\times C|$. If you also know that there is no surjection between $A$ and $B$, then again you are going to have two different cases, depending on whether $C$ is "really big" (compared to $A$ and $B$) or not.
Problem 3. The set $A^B$ is the set of all functions from $B$ to $A$. So $\mathbb{Q}^{\mathbb{N}}$ is the set of all functions from $\mathbb{N}$ to $\mathbb{Q}$; this is the set of all rational sequences (sequences, each term a rational number). The set $\{0,1\}^{\mathbb{N}}$ is the set of all binary sequences (sequences, each term either $0$ or $1$). Hint. Think about binary expansion of real numbers between $0$ and $1$, or even better, the ternary expansion of the elements of the Cantor set to deal with $\{0,1\}^{\mathbb{N}}$. For $\{0,1\}^*$, can you count how many sequences of length $n$ there are, for each $n$? Add them all up then. For $\mathbb{Q}^{\mathbb{N}}$, can you exhibit at least as many sequences as there are real numbers?
Problem 4. You are comparing sets by inclusion. We say set $A$ is "smaller than or equal to" set $B$ if and only if $A\subseteq B$. So, for example, the set $\{3\}$ is smaller than or equal to the set $\{1,2,3,4,5\}$, but not smaller than the set $\{1,2\}$. When using this order, note that it is possible for you to have two sets, neither of which is smaller than or equal to the other (for example, neither of $\{3\}$ and $\{1,2\}$ is smaller than the other; they are incomparable). The maximum among the sets you are given would be a set that is greater than or equal (contains) all of the sets you are given; a minimum would be a set that is smaller than or equal (is contained in) all of the sets you are given. They may or may not exist. "Greatest" is the same as "maximum" in this context; "smallest" the same as minimum.
Problem 5. An equivalence relation is a collection of ordered pairs that is reflexive, symmetric, and transitive. A partial order is a collection of ordered pairs that is reflexive, *anti*symmetric, and transitive.
So the question is: how many equivalence relations on $\mathbb{N}$ are also antisymmetric? Or: how many relations are all of reflexive, transitive, symmetric, and antisymmetric? Think about what having both symmetry and antisymmetry means.