How do you measure proton's spin? I've probably read it somewhere in Sakurai but I cannot recall it at the moment. So how does one really measure the proton's spin? I mean the proton's spin and not its constituents. 
Do you measure it using a Stern-Gerlach type of setup? 
How about atoms and other stuff, how is their spin measured?
 A: Spin can be measured as you say by the Stern Gerlach experiment for individual particles.
Spin for atoms can be found from measuring electromagnetic spectra of transitions between energy levels and fitting/assigning a solution of the potential problem which identifies the spin state for us.
As the answer by Dwin says there are specific energy levels that can be excited in nuclei and fitted with the appropriate model to identify the spin.
In accelerator experiments the spin of resonances can be determined from their decay products, fitting their angular distributions . I have provided some links in the answer to a similar quiestion.
A: Put them in an NMR (aka MRI) machine. NMR = "nuclear magnetic resonance" and hte nucleus most often measured is the hydrogen nucleus, i.e. the proton. Basically you apply changing axial and inhomogeneous magnetic field components and measure the microwave radiation that is emitted from transitions from excited spin states to lower energy states (induced by the axial field). To do NMR measurements you need a non-zero total spin number, so deuterium or He_4 nuclei are no good.
