I have looked up all three in my dictionary and found they roughly meant the same thing. They all meant completing something. How do they change in meaning by adding off or up after finish? Dictionary did not really help me to fully grasp the differences. Does finish off mean that you do it long time and finally complete it or does it involve a series of steps to finally complete that task? How and when do you people use finish or finish up or finish off?
They do mean essentially the same thing, the up and off are mostly added for emphasis
Finishing off or up is generally used near the end of a fairly long task, or when specifically talking about the last portion of a task.
eg if I've got nearly a whole drink left, a friend is most likely to tell me
We'll finish our drinks, Jon, and get going
Whereas if I've got a couple of mouthfuls left he may say.
Finish your drink off, Jon, we're all waiting for you.
The latter both emphasizes the sentence (because my friends are waiting for me) and references the fact I'm nearly finished. The only real difference they apply is the emphasis - usually (but not always) requesting that the end of the task is expedited, or hinting that the task is taking too long.
They both tend to be added to long tasks, or tasks which are taking longer than normal, but this is by no means a rule. Similarly you'd tend not to add either to a short or quick task, as there's no need to emphasize... again, this isn't a rule, just a description of the most common usages.
Overall, off or up are virtually never necessary, they're colloquialisms added to informal speech. That said, they've made their way into formal speech too, and are fairly common.
If in doubt, you'll almost always be safe using "finish", but you'll also almost never be wrong adding "up" or "off" to the end.
(Note: the one caveat to this is that to "finish" someone would generally imply either killing them or knocking them out, if talking about a violent situation. "Finishing someone off", however, may imply something of an entirely different physical nature which I won't describe on StackExchange, but I'm sure you can guess...)