I have heard a lot about dark souls 2 and want to get in that game now. So can I start with Dark souls 2 straightaway or first I should complete Demon's Souls and Dark Souls. How important and connected the story is?
The story isn't really connected with Dark Souls 1 other than it's the 'same universe'. At least, not that I noticed.
Both games have huge amounts of lore you can dig up by finding things, and reading online where you can't be bothered to find things ;) - However if you've played DS1 you'll have a good idea of the combat and other ideas in DS2. But this doesn't mean you can't just jump in with DS2, it'll just take you a bit longer to pick up the games style.
(I've not played Demon Souls so can't comment about that).
There really is no necessity to play 1 before 2, the story is not linked in any way other than a "repeating cycle".
That being said however, there are references in Dark Souls 2 that link back to Dark Souls 1. Eg, you can find the orginal Lords of the Flame in DkS 1, and only references to them in 2.
The storyline and Lore can be a bit vague if you're not really looking, but IMO, I would suggest playing them in order, just to get a proper understanding of the universe. (Also, I would suggest playing 1 first, otherwise you'll likely get annoyed by the clunky controls compared to 2).
In regards to Demon's Souls, there is no relation whatsoever to Dark Souls.
DS1 and 2 both tell their story in an unusual way: rather than massive lore dumps like most RPGs they expect you to piece the story together yourself from environmental and item description cues. Some of these can be very, very subtle (Vaatividya and others have produced excellent interpretations), and you'll have to be very observant to understand in any kind of detail what's going on by the time you end DS1. Items like the Cloranthy ring show up in both, but connections seem to be few and tenuous.
The mechanics are quite similar - DS2 has more polished combat mechanics, but more inventory clutter IMO. If you learn to operate one you know 90% of what you need to know to get into the swing of the other.
I started with DS1 because of similar concerns, and ended up using the wikis and YouTube to learn the lore as I went along. They complement each other beautifully, and I think has made the introduction to DS2 more pleasant as well. YMMV.
If you are on the PC, I would highly recommend going straight into Dark Souls 2.
The PC version of Dark Souls 1 is an extremely poorly done port, and you basically must setup a game console control to play it. You need to install a bunch of mods to compensate for the poor mouse control if you want to stick with Mouse and Keyboard.
I did get through DS1 on Mouse and Keyboard, but I tried replaying it after DS2, and it was just really bad.