On the edge of World War
To advance to the next era, the World Wars era, you have to fulfill one requirement - your support on the island has to be over 51%.
One way to increase support is to try to meet the needs of your citizens by for example building Taverns, Catholic Missions, as well as workplaces and houses.
Creating a Constitution will increase support and get you closer to declaring independence, so it's good to invest in scientific development. You should build a Library at the very beginning. However, don't hire scientists from abroad, because it's extremely expensive. After a few ships arrive at your shores and after you complete a couple of missions, you should be able to find such people on your island.
From time to time, missions for revolutionaries will appear. For completing them, you will be rewarded with money or with three educated revolutionaries.
Ice and snow. Cold war
You enter the third era, which is the Cold War era, when you sign all the treaties with both sides of conflict.
If the relationship is warm, or more than that, the treaty will be signed without hesitation. Otherwise, you should expect some conditions to fulfill, or even an invasion.
Manage your trade routes well. They influence the relations with other countries.
Another way to improve the relations are Embassies, which can soothe the anger of Hitler or Churchill.
There are some missions that exclude one another. If we see two exclamation marks over the palace, it is possible that they indicate such a mission. We cannot serve both sides in those missions.
Finally, other small things that influence the attitude of other countries towards you are laws and articles of constitution. For example, Americans will like "Immigrants Nation" and the Axis will be fond of "Professional Army"
To step into the last era, the Modern Times era, you will need two things: Space Program (peaceful way) or Nuclear Program (less peaceful way).
The basic thing that is necessary to build those is scientific development. Both buildings are available after discovering appropriate technologies - "Hostile Atom" and "Space".
The following buildings will help you to reach this goal quickly: Libraries, Colleges (they have to be upgraded) and Observatories. Each one of them brings a lot of development points. If you increase their budget to the maximum and find a manager with "Tycoon", the research process will speed up considerably.