In the PC version of the game, the Quest Log can be seen on the left side of the screen as shown in the screenshot below:
As you can see, there are different colors of the quest backgrounds. Each color has its meaning.
The quest that you need to complete in order to proceed in the game, are the ones with the golden (or yellow?) background. By completing these quests, you´re proceeding in the main questline. These quests are a must if you want to progress within the game.
Then, there are side quests. Those are the ones with the blue background (the last one on the screenshot). Most of the time, you don´t have to complete these in order to proceed in the main questline. There are a few exceptions though, therefore, if you´re not able to find more quests according to the steps mentioned in the links below, you might want to complete some of your side quests first.
The quests with the red background, as the first one on the screen, are the event quests. These need to be completed in order to complete special events introduced in the game. However, you don´t have to complete these in order to proceed in the main questline of the game.
More information about the questlines as well as the specific quests, that need to be completed for each region can be found here.
What to do, when you run out of quests?
In general, quests are being given by people in the City or by gods themselves in the Temples.
In case you´re having issues finding a new quest in Egypt, please check this article.
If you´re having issues finding new quests in Scandinavia, please see this.
In order to find more quests in China, please look for a piece of advice here.
To be able to get more quests in Atlantis, make sure to visit this link.
In order to get more quest in Greece, please check the following article.
And if you´re in need of more quests in the New World, please check the steps here.