Given an elliptic curve, how do I calculate the order of the points manually, when we don't know about the curve's points? So basically all I can do is 


use Lagrange's Theorem and figure which factors of the group order are in line, then 
start trying each of these using the Double-and-Add-Algorithm until I get $\mathcal{O}$?


This takes forever if I do it by hand. Is there any easier way to find the order of all points? or find all the points on the curve?
Okay after some research: There is no shortcut. If you do it manually you need to proceed as suggested above. Realistically this is possible only for very small curves because of the complexity.