If I remember I think I did run through the tutorials but I pretty much just jumped in and started playing. I did start out in smaller galaxies and at beginner level but I didn't bother to finish the first game I played or even the 2nd or 3rd. I have never played a campaign.
At beginner the AI really doesn't do much of anything but it's as good a place to start as any just to get the idea of how to move around and colonize planets. Note that at beginner you get bonuses and the AI is pretty well crippled.
Don't dwell too long at a level once you become a bit more confident because it will let you develop bad habits. Once you get up to tough or thereabouts the AI is fully functional and you no longer get bonuses and the real game begins. This is the point you really start learning. The AI aggressively competes for planets and mining resources and most likely will kick your butt rather soundly, but keep at it and you will progress.
There's all different ways to play the game and strategy is a common topic of conversation but it's really hard to answer the inevitable question of what is the best strategy. I really don't think there is a best strategy, there's only the best strategy for you at this moment in time in your development that best fits with your developing style.
The obvious path to victory is to colonize more planets and develop them quicker than the AI does, but as you move up in difficulty this gets harder and harder to do. There are ways using diplomacy, of paying one AI to make war with another AI and then try to pick off the pieces. There are strategies that focus on one concentrated area of research to gain a critical advantage in that area and then use that to conquer one of the AI's. Generally speaking once you've conquered one AI you're usually well on your path to victory.
There are just so many methods to play this game it's not really possible to begin to discuss them all within the confines of a thread like this and even if there were you're not going to understand most of them until you get into the game and start gaining some experience. Once you do start to gain some experience then you'll need to ask many questions here and you'll get many good answers some of which will work for you and some of which won't.
The best questions to ask are the ones that are the most specific and detailed. The "big picture" questions are really the hardest to answer, the little details are most often much easier to answer. "Big picture" questions often get long rambling responses with not much detail (very much like this one).
Anyway the only way to really get started is to dive on in.