To expand, you have to level up by collecting Experience and have the technology Underworld Control researched.
To get your first expansion dungeon, you will need to complete enough of the quests to reach level 10. You do not gain experience over time until you reach level 10. The only way to get experience before you are level 10 is to do the quests. Once you reach level 10, then your dungeons will produce experience over time based on the rooms you have built. You can still complete the quests after level 10 in order to gain the experience rewards to advance faster.
Players can, at most, own one dungeon per 10 Overlord Levels. As you only having a few goblins to do surveys when you only have your starter dungeon, it is recommended you survey your mountain completely and any mountain within a few tiles before you reach level 10, so that on hitting level 10 you can take the best one in the area. The most basic dungeon you take should include both Iron and Crystal at minimum or you'll spend a lot of time shipping resources into the dungeon to make room upgrades. A preferred dungeon will also include Primordial Earth, so that you can upgrade your rooms even further as well as make Steel Ingots (which are the main component of a lot of needed furniture and room upgrades) and Primordial Water, which is used to make Crystite (a main component in quite a few higher-end room upgrades and furniture). Any dungeon that includes Iron, Crystal, Primordial Earth or Primordial Water and any additional resources besides gold (they all have gold) would be a superior 2nd dungeon that you should take right away. Do not concern yourself with taking dungeons with only higher materials for your second dungeon or you'll seriously harm your progression.
Now that you have two dungeons, you can start the scouting for a third dungeon for when you hit level 20. You'll have about two weeks' time before you hit level 20, so you can search a lot of mountains to find some nice dungeons. Once you've scouted a dungeon, the results will be recorded in your log book. The log book is found in the overworld screen on the tabs to the right. It allows you to quickly search for a specific mine node among the dungeon locations you have already scouted. This tool will be very valuable as you pick your 3rd, 4th, 5th and so on. That great dungeon full of high level resources may not help you on your 2nd or 3rd dungeon but you want to remember where it is when you reach level 30 or 40 without the hassle of clicking through dozens and dozens of mountains trying to find it again.
It is also recommended that your third dungeon also has Iron and Crystal in it along with other resources in order to avoid excess shipping of resources. Iron and Crystal are used to create many units and rooms, and will become hard to keep in supply if you don't have mines in each of your first three dungeons. By level 30 to 40 you can start picking dungeons without Iron and Crystal. You will be raiding more often at these levels and building up Iron and Crystal that way. If you raid inactive dungeons you can gather a significant amount of Iron and Crystal very quickly, as most inactive starter dungeons usually have 750 iron and 750 crystal, perhaps more, sitting there for the taking every day. Raiding settlements also builds up Iron/Crystal pretty fast. By this level you will have better research skills that increase mining rates, higher level mines, better furniture to increase mining rate, and additional goblins, so the rate at which you produce Iron and Crystal from the same dungeon will increase significantly.
Once you reach level 20 you might want to start looking towards specializing your dungeons for when you hit level 30, so that you can get the most out of your dungeon tiles. A level 5 room takes up a lot of space (25 tiles) so you can't have a level 5 room of every type in each dungeon.