Well I'm a newbie at this game.

I encountered this dragon right when I touched down from the horse carriage at Windhelm. So yeah this frost dragon appeared, I killed it, absorbed its soul but I didn't learn any shout. Is this normal?
How do I gain more shouts then..? I have 1 shout at the moment, unrelenting force, gotten from killing the dragon while doing main quest near Whiterun.

You have to learn the dragon words. Have fun locating them.

You learn words at different dragon shrines located throughout Skyrim, and you use up a dragon soul to "unlock" it so it becomes usable.

You learn Shouts from word walls (mostly found in dungeons), and then need the souls of a dragon to unlock them.

:ninja: 'd

You learn words of power from word walls. They look like this: http://images1.wikia.nocookie.net/__cb20111115070844/elderscrolls/images/thumb/7/79/5819446956_7090e3d5f3.jpg/300px-5819446956_7090e3d5f3.jpg
As you approach them, you will begin to hear chanting

Words are learned from word walls, souls are just used to unlock the shouts you've learned.

You have to visit Word Walls scattered around Skyrim and learn the word off the wall. And then you have to spend a dragon soul to unlock the word.

A word of advice: Don't spend souls on unlocking shouts you don't expect to use. Save 'em 'til you need 'em.

yes as the above have said, you only get dragon souls from killing dragons.

you then use these souls to 'power' your words that you learn from the word walls.

so basically, you learn the words from the walls, but you can't use them until you kill a dragon, absorb its soul, then use that soul to unlock your word

How did I get this "unrelenting roar" then...?

Well they had to give you something at the start to whet your appetite at the beginning of the game.

Word wall in bleak falls barrow.

The thing is i've touched the word wall first before I killed the dragon at whiterun. The optional quest even told me to do the Shout in front of the ally guards. Meaning I got the word wall first then the dragon soul...?

How do I check how many dragon souls I have?

Magic menu > Shouts. Bottom right corner shows you how many dragon souls you have left.

You can learn words or absorb souls in any order. You can roam around absorbing infinity souls and not know a single word to use them on if you wanted to.

OP, this video should explain things http://www.youtube.com/watch?v=2QamU4-8NUw