well the classical simple method of starting cultures with fixed numbers in same conditions ,then detaching cells and counting a sample every 3 days by classical living dead methylene blue stain count on a hemocytometer  is  an easy simple and straight foreword method with limited cost although many advances in high through output real time culture dish photography is a very interesting new method yet expensive and requiring many equipments 

You could use a biochemical assay such as MTT or Alamar Blue which rely on the ability of cells to metabolise the compounds in these reagents. The more the readout, the more the viability/number of cells. The readout (absorbance/fluoroscence) can then be compared to a standard curve obtained from a range of known numbers of cells to estimate cell numbers in your test wells at each time point, thus giving you a rate of proliferation when compared to your starting cell numbers. 

The approach suggested by Ahmad is also known "3T3 assay" and could also be quantified manually. I usually had more biological replicas (such as 3-5 wells) for each condition, to make a sort of statistical analysis. An other possibility is to perform a colony assay, by plating cells at very low density (for instance 1000 in a 10-cm plate) and let them grow for a couple of weeks. When colonies are starting to cover the plate, you can fix and color it by Crystal Violet solution. It is not quantitative, but gives you an idea of the proliferation capacity of analyzed cells.

Depends on the purpose of your experimentations.

In addition, what the scientists already explained above, I suggest looking at cell proliferation reagent WST-1. Further more, you could check the expression levels of cell cycle related transcripts (and protein), and cell proliferation related transcripts as well.

With all due respect, cell proliferation rate is not DNA synthesis.To measure cell proliferation in culture conditions, one is supposed to do nuclei counts in a particle counter. For this purpose, cells (nuclei) should be counted in a Coulter counter-type devise as a function of time.  .