Here are some helpful directions:
The Spire can be reached by first going to Tau Dewa, Azure Sector. Once there, use the transwarp conduit to go to the Dyson Sphere. When you are in the Sphere, travel to the northernmost border. That will bring up the tower window that asks if you would like to go to the fleet spire.
The Dilithium Mine is in the Beta Ursae, Bajor sector, SW of Deep space 9. It shows up on the map list...
The Embassy is also located in Tau Dewa, Nimbus sector. Go to New Romulus. Once you get to New Romulus you will have the option to visit the Embassy or go to New Romulus- take the Embassy of course!
The Star Base is in Eta Eridani, Aldebaren sector. Located in the southernmost area...
Once we reach a high enough Tier, we can get Transwarp abilities to get quickly to any of these locations quickly-and freely.