I am wondering what would be the easiest way for me to go ahead about. I do not need email since I have email stuff from paid hosting. So what I am interested in is self hosted calendar and address book type of things.
The easiest thing for you to do would be to find a hosted service. Zimbra OSE, out of the box, does not support syncing calendar/contacts with smart phones.
There are add-ons available, such as Z-Push and Funambol that would allow it, but require an additional machine (Or virtual machine if you're using Virtualization).
Ben Franklin quote:
"Those who would give up Essential Liberty to purchase a little Temporary Safety, deserve neither Liberty nor Safety."