You do this in a wide variety of means. You can expand your nation's borders by taking over the provinces of another nation. There are two big ways of doing this:
- War: You can declare war on an enemy nation and take it provinces by force. Your armies will have to occupy the enemy provinces, doing so will increase the war score for you and your allies. The higher your war score, the more provinces you can claim. Having a war score of 100% allows you to fully annex an enemy nation.
- Royal Marriages: Royal marriages a peaceful way of expanding your territory. By engaging in a marriage your bond the royal lines of both nations. If one of the nations involved falls without a legal heir to the throne, the other nation can claim the throne. At this point you can either wait till the current ruler dies without a legal heir, in which case you will inherit the throne, or you can declare war on the enemy nation and force them to make a persional union with you. In which case you can integrate the nation after a certain period of time.
- Colonization: Once the quest for a new land begins, most (western) nations that border to a sea will be able to start sending out colonists to foreign lands. If your colonization attempt is succesful, the oversea province will join your nation and fall under your influence. However, you need 1000 people for the province to become self-sustaining, this means that sending about 10 colonists to a province will make it self-sustainable. These people also generate over time, but colony maintenance is pretty high, so it's best to find a good balance in this.
Keep an eye out for inflation! Inflation is influenced by sliding your treasury slider, this increases or decreases your gold income to the treasury, which is basically the gold you get to play with. Every % of inflation makes it more costely to sustain your nation (1% of inflation makes troops 1% more expensive, etc.), but it is a great way to fill up your treasury. For new players I recommend keeping your inflation at 0% by hiring a suited advisor or using the national bank as national idea (-0.10% inflation yearly). This allows you to fill up your treasury without gaining inflation.
Players who have a little more experience can use inflation to fill up the treasury early game, and get rid of the inflation later on. It's a good way to start out and build up your nation and maximize your incomes.
Important: changing the treasury slider also influences the amount of gold you are investing in research and technology. This is a crucial aspect of the game as it determines what you are able to do as a nation. A higher land technology allows for better troops, a higher trade technology gives more trading opportunities etc. If you fall too far behind on your neighbors, it might become an issue when it comes to a power struggle.
The goal of the game is basically just surviving, but it provides a richer gameplay experience if you set some goals for yourself (e.g. conquer Northern Europe). Most nations in the game get some sort of missions, which usually are historical facts which can be persued for a bonus reward to your nation.
Tired of playing against the AI? No problem, with the Divine Wind expansion, EU3 features a multiplayer mode where you can fight against or alongside your friends. Which provides a unique game experience every time.
If you're a fan of strategy games, but haven't played grand-strategy before. This is a relatively easy game to get into. Paradox has made many games which are a lot harder to play than Europa Universalis (like Victoria). After playing the ingame tutorial and reading the basic wiki pages, you should have a very good idea of how to play the game.
I really recommend picking up this game, it'll provide hours of unique gameplay in an always changing political world.