What is the proper way to prepare a solution from hydrate?
Originally I thought that all I have to do to figure how much $\ce{Ca(NO3)2}$ to weigh is
  
  
  and then just add  $10~ \mathrm{mL}$  water.


To properly make an aqueous solution of a specified concentration you should dissolve the desired amount of material in about $80\%$ ($8~ \mathrm{mL}$) the desired volume of water. Then dilute to the final solution volume (add water until the solution volume is $10~ \mathrm{mL}$).  


  How can I verify that I got $\mathrm{1~M}$ $\ce{Ca(NO3)2}$ as opposed to $\mathrm{1~M}$ $\ce{Ca(NO3)2*4H2O}$? 


Hydrates are for the solid state.  You do not have "hydrates" in the aqueous phase.  Therefore the two solutions are identical and indistinguishable.