It really depends from which version to which. From my experience with WP: It can go flawlessly; it can also turn out to be a terribly cumbersome work, depending on what plug-ins are running and what else has been done with the installation.
Anyway, I'm pretty sure there are extensive migration guides on the Wordpress site - reading through, and working with those should be your first order of business.
A few general points:
How should i take backup of all things?
Do a full backup of all the site's folders through FTP, and make a dump of all databases used (it's usually one database). You can use phpMyAdmin's "Export" function for that.
Should i take backup of database also?
Yes, definitely.
How to upgraded modified plugins without loosing functionality?
That is impossible to answer without knowing the plugins. It can range from "will work flawlessly" to "awful mess". Best ask the person who modified the plugins.