Exploring the galaxy is one of the basic actions during the first few hours of the game. The system that allows you to scan planets and discover their wealth was made in a such way that even in the later parts of your game you can still go back to exploring the galaxy. In order to do that you can use the special type of ship that has Exploration role (those with magnifying glass icon). Those ships are weak but, usually, they have a few module slots. A unique module, available only for this type of ships, are probes. Hero ships can also be equipped with that module. Although the upgrade is installed immediately it costs Dust and in the beginning of the game you don't have a lot of the currency. The ships can move like other fleets which means that you can send them into unknown systems by using the connections between them.
To use a probe you must press the special button in fleet's menu. When you select a group with an exploration ship then on the left you can see the Launch Probes option. Thanks to that you can send a probe in the direction that you currently can't or you don't want to move. Usually it is a good idea to send a probe in a direction that you can't currently move to or you want to be sure whether it is safe or not to move there. It is worth to build a few exploration ships and send them in small groups. Thanks to that it is easier for them to defend from pirates and they will have more probes. Later you can get access to better probe modules.
Curiosities that can be found on planets, are randomly generated events (similar to, for example, Tribal Villages in Civilization 6). From the game's screen you can easily locate those places. Under a system's name you can find a number of planets. Those planets that are pulsating have at least one Curiosity to examine. In order to do that you need a ship with probes. You can now select Start Expedition option or you can zoom in until you see all planets in a system. Next to names and statistics of planets you can see Curiosity icons. When you click on the button you send a probe and on the screen you can see a window informing you about results. There are a few different Curiosities - sometimes they give you different effects but your chances to find something interesting may vary.
After examining a Curiosity you can get positive or negative consequences. In most of the cases you get deposits of strategic or luxury resources and from time to time you can get a small amount of one of the FIDSI resources. You can also meet someone from other civilization that will join you (a civilian ship is sent to the nearest system that has enough living space). You can find good or negative anomalies on a planet. Sometimes, you can receive wisdom that speeds up your research on one technology or you can get a unique module for your ships (usually better than most of those that you currently have or can research). In some cases you may find nothing and sometimes you can find something unpleasant, like being attacked by pirates.
Some Curiosities requires higher level of Curiosity Detection Level or Curiosity Expedition Power. You can often see a Curiosity but you won't be able to send your people there. An icon for such Curiosity is darker and has a lock. This is a sign for you that you must level up said values. The only way to do that is to research certain technologies. Besides that if you unlock a new technology tree and you get the access to another Era of Science and Exploration you can get a better chance finding rare resources than you usually have. You should invest equally in that domain of research and in exploration fleet if you want to discover all secrets of the galaxy.