Is there some sort of equation/formula that can be used to find the exact values of $m$ and $b$ in $y=mx+b$ of any data points for the line of best fit? I want to be able to do this manually, not with graphing tools.
I know with $2$ points, $y_1-y_0\over x_1-x_0$ will satisfy, but with $3$+ points, there must be some definite method of interpolation, no?
In order to calculate the line of best fit you have to minimize the quantity $$\sum_i^N [y_i-mx_i-q]^2$$ with respect to the two parameters $m$ and $q$. The $y_i$ and $x_i $ are yours data. The first equation is $$2\sum_i^N[y_i-mx_i-q](-x_i)=0$$ while the second $$2\sum_i^N[y_i-mx_i-q](-1)=0.$$ Try to isolate $m$ and $q$ and remember to use $\sum_i^Nq=qN$ (relevant quantities are $\sum_i^nx_iy_i $, $\sum_i^nx_i $ and $\sum_i^nx_i^2$). This is the general method used also for a general fit given by $y=f_{a,b,c,\dots}(x)$ where $a,b,c,\dots$ are your parameters. Minimizing the function with respect to these parameters and setting to zero each times give you a set of equations in number equals to the number of parameters.