I am thinking to sync my nokia c5 03 calender and contacts to google server, just need a free service.
easiest way to do so is over the active sync. Google provides you a good working solution!
In you manual, you'll find out how to configure an Exchange Server on your phone, Nokia calls it "Mail for Exchange".
It supports to sync following information:
Mails (including Push)
Here you will find the instructions by google: support.google.com/mobile/bin/answer.py?hl=en&answer=147951
Here is a link to the nokia manual of your phone: nds1.nokia.com/phones/files/guides/Nokia_C5-03_UG_en.pdf
Go on page 43 of the Nokia manual.