In what instance would r, R, and β be the same? I understand that r tells us the strength of the linear relationship between two variables, R shows how closely two variables, and β shows which dependent variables would change if we change the independent variable by one unit while other independent variables constant (multiple regression?). 
I'm not able to grasp the bigger picture and relationship between these variables, perhaps I am not connecting parts of the equations. Any help would be greatly appreciated.
 A: Pearson's $r$ correlation coefficient is computed as the covariance of two variables, divided by the product of their standard deviations. It is bounded between -1 and 1.
$$
r_{x,y}=\frac{\text{cov}(x,y)}{\sigma_x \sigma_y}
$$
Now, let say you estimates the simple linear regression model
$$
y_i=\beta_o+\beta_1 x_i + \eta_i\\
\eta \sim \mathcal{N}(0,\sigma_\eta^2)
$$
In this model $\beta_1$ indicates by how much $y$ is expected to change following a unitary increase in the independent variable $x$. The linear slope $\beta_1$ that minimizes the squared error (the squared differences between the actual and predicted values of $y$) is computed as 
$$
\begin{align}
\beta_1&=\frac{\text{cov}(x,y)}{\sigma_x^2}\\
&=\frac{\text{cov}(x,y)}{\sigma_x \sigma_y} \cdot \frac{\sigma_y}{\sigma_x}\\
&= r_{x,y}\cdot \frac{\sigma_y}{\sigma_x}
\end{align}
$$
It follows that if $\sigma_y=\sigma_x$ then $\beta_1=r_{x,y}$. Indeed if the two variables are standardized, so that both their standard deviation are equal to 1, then the estimate of the slope $\beta_1$ will correspond exactly to the correlation coefficient $r_{x,y}$
I assume that with $R$ you intend the coefficient of determination $R^2$. This indicate the proportion of variance explained by a model, and is calculated as the ration between the explained and total sum of squares
$$
R^2 = \frac{\sum_i(\hat y_i - \bar y)^2}{\sum_i(y_i - \bar y)^2}
$$
where $\hat y_i=\beta_o+\beta_1 x_i$ are the predicted values of $y$ (and $\bar y$ is the mean of $y$). It can be shown (see wikipedia for the derivation) that $R^2$ is the square of the correlation coefficient between the predicted and observed values, that is
$$
r_{y, \hat y}= \sqrt{\frac{\sum_i(\hat y_i - \bar y)^2}{\sum_i(y_i - \bar y)^2}}=\sqrt{R^2}
$$
