To be neat, I want to do something like this, but with my model:
(http://docs.autodesk.com/3DSMAX/15/ENU/3ds-Max-Help/images/GUID-9585BFA1-A0E6-412B-B845-5CEC60F138CB-low.png)
You probably won't want to do this, since artificially smoothing the model by inflating the poly count will most certainly cause lag, but using a Turbo or Mesh Smooth modifier will accomplish what you're looking for. I'd recommend adding geometry manually and building it up until you reach a level of smoothness that you're satisfied with.