Do I need to begin the main quest aswell? In the quest where I'm to deliver an axe to the Jarl of Whiterun there's no option to just give it to him, I can only carry on with the main quest.

In order to play either side of the Civil War storyline, you will need to advance the main quest up through the part where you kill the first dragon outside of Whiterun. It doesn't matter who you sends you to Balgruuf, Ulfric or Tullius. He ain't hearin' no war-related nonsense until you take care of his dragon problems.

the axe for the jarl of Whiterun from Ulfric in Windhelm IS part of the civil war quest. And part either Imperial or Stormcloak. If the Jarl of Whiterun gave the axe to you to give back to Ulfric then you already are beyond switching to the Stormcloak side.

But if Ulfric gave the axe to you... have you done your newbie kill an ice wraith to qualify to be a stormcloak yet? You can completely finish the CW quest without the main quest.

Its the mage college you need to at least start in order to finish the MQ.

And after that I'm good to complete the entire civil war questline?

The only part of the main quest you have to do is Dragon Rising (and anything before it).

Yes, that is the only place where they absolutely must intersect in every playthrough. After that you can complete the CW without advancing any further in the main quest.

However, you will NOT be able to complete the main quest while the CW is ongoing. If you continue doing main quest stuff simultaneously you will eventually reach a point where the people you need will not help you because of the war and you'll have to negotiate a truce. After that you will have to finish the main quest first and then resume the civil war storyline afterwards.

If your intention is to play through the Civil War and ignore the rest of the MQ for the duration, though, you'll be fine.

As a side note, if you are concerned about having to deal with random dragons spawning while you are in fact ignoring most of the main quest, do the bit you have to do in Whiterun and leave it at that. Some players say that if you don't get to a certain later part of the MQ, random dragons don't start spawning in your game. I can't confirm this but some folks swear by it. There will still be dragons in known dragon lairs, guarding word walls and stuff, but supposedly they won't just be popping up all over the place. If you stay away from their lairs you don't really have to deal with them.

Dragons attack me all the time. Two days ago one attacked me by the stables in Windhelm and another on the plaines of Whiterun. And the 2nd dragon killed 2 mammoths to boot while it was straifing me.

I logged a game save just before joining one of the sides. That way I can give both sides a shot since I'm a little indifferent on who to join.

Not really. If you know where to find the person you're supposed to look for you can go straight there (can of course be hard to motivate how your character knows, if you RP). And even if you go to the College, you can just ignore the quest that starts for the rest of the game.

Same with Thieves Guild. You don't have to ask Brynjolf, as Delphine suggests. Keerava will also direct you to the next person (and to me, it seems much more logical for my non-thieving characters to talk to the innkeeper rather than that suspicious guy who sells dubious potions and accuses me of being a criminal).

To complete the "message to whiterun" you have to finish the "bleak falls barrow" and "dragon rising" quests. I would recommend to do the main quest first though because you learn a lout of good shouts from dungeons and storyline, plus you will level up more.

Where are you in the Main Quest? If you have progressed beyond the point where

then, yes, you will definitely have random dragons in addition to the ones guarding word walls and known lairs. But some people say that random dragons do not spawn or attack them in other locations before that part of the main quest is complete.