Understanding gravity
The magnitude of the force of gravity between two bodies is proportional to the product of their masses:
$$F=G\frac{m_1m_2}{r^2}$$
This doesn't change depending on which body you're applying the force to, i.e. if you interchange the masses. The magnitude is the same.

What does change is the direction of the force. Force is a vector quantity, denoted as $\vec{F}$ or $\mathbf{F}$. If we write the equation for gravity using proper vector notation, we have
$$\mathbf{F}=G\frac{m_1m_2}{|\mathbf{r}_1-\mathbf{r_2}|^2}\frac{\mathbf{r_1}-\mathbf{r_2}}{|\mathbf{r_1}-\mathbf{r_2}|}$$
Here, the positions of the objects are represented by vectors, $\mathbf{r}_1$ and $\mathbf{r_2}$. Additionally, $|\mathbf{x}|$ denotes the norm of a vector $\mathbf{x}$ - its magnitude.

Now, if you interchange the masses, the direction of the force changes, although $|\mathbf{r_1}-\mathbf{r_2}|=|\mathbf{r_2}-\mathbf{r_1}|$, because this refers to the magnitude of the vectors. So the force applied on one object is the opposite of the force applied on the other object. This is Newton's third law.

The acceleration is more interesting. The force on object $1$ due to gravity is
$$F_1=m_1g_1$$
Here,
$$g_1=\frac{Gm_2}{r^2}$$
where $m_2$ is the other mass. This should tell you that $g_1\neq g_2$, except when $m_1=m_2$.

I'm not an expert in general relativity, but I do know that it describes how spacetime curves due to the presence of one body. The solution to the Einstein Field Equations, the metric, is different for different bodies, because one piece of it, the stress-energy tensor, is different for objects of different mass/energy/etc.