I know this is sort of a basic question, but I don't completely understand the following. Let $A$ and $B$ be two problems. If I take one instance $a$ of $A$ and one instance $b$ of $B$, and show the following. Solving $a$ implies solving $b$, and solving $b$ implies solving $a$. Am I showing that $A$ reduces to $B$, $B$ reduces to $A$ or both? How are $A$ and $B$ related using the $\leq_P$ and $=_P$ notation?
Neither -- you can't show a reduction by starting out with an instance of $A$ and an instance of $B$. Instead, a reduction has to be a way of transforming instances of $A$ into instances of $B$, and vice versa.
A reduction from $A$ to $B$ means you give a way of transforming any instance $a$ of $A$ to an instance $f(a)$ of $B$, such that $a$ is a yes-instance of $A$ if and only if $f(a)$ is a yes-instance of $B$. Here $f$ is the reduction, or transformation. Notice that $f(a) = b$ might not just be any instance of $B$, but it might be a very particular instance constructed by our transformation.
For example: suppose $A$ is the problem of determining if a number is even, and $B$ is the problem of determining if a number is a multiple of 10. Then to give a reduction from $A$ to $B$ we could start with a number $n$, and multiply it by $5$. Our reduction is:
f(n) = n * 5
This is a valid reduction because
n is an even if and only if
n * 5 is a multiple of 10. But, it doesn't construct every possible instance of $B$; only multiples of $5$ are actually constructed as instances of $B$, not all integers.