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 in order 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.
You should gain 10 new levels roughly every few weeks depending on your room upgrades. The fastest way to gain experience is to first upgrade all vaults to level 2, then all vaults to level 3, then upgrade your Dens to level 3 (this doesn't give additional experience but does add one more goblin to the dungeon which will help you mine more resources).
Eventually, as you have the research technologies to make the needed items, upgrade all mines to level 5, then all vaults to level 4. This will be about as far as you will likely get under level 30, unless you specialize dungeons, due to tile limitations. Once you get the technology Stone Binding you will have additional tiles and be able to upgrade more rooms to level 5. For each non-Vault Room that is currently at level 5, an additional 1 XP/hour is gained.
Players can, at most, own one dungeon per 10 Overlord Levels. Due to 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, then on hitting level 10 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. Any dungeon that includes Iron, Crystal, Primordial Earth 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. You will need significant amounts of Primordial Earth at level 10, as well as iron and crystal.
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. You should keep track of any good dungeons you find by recording the mountain name, coordinates and what mines are inside. This way before you pick your new dungeon location you can make a decision based on that list. That list 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 for your third dungeon that you also pick one that has Iron and Crystal in it along with other resources in order to avoid excess shipping of resources. Iron and Crystal will become hard to keep in supply if you don't have mines in each of your first three dungeons as you use a lot of it to build your troops and rooms. 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 through raiding. 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 sitting there for the taking every day, if not more. Raiding settlements also builds up iron/crystal pretty fast as well. 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, and having the same level 4 rooms in each dungeon just limits your experience growth as only level 5 rooms give experience.