To access a new realm you must unlock it, unlocking a realm is typically done by speaking to an NPC (quest) who will then show you how to get to the new realm.
New realms can usually be unlocked at 100 level intervals starting at level 95.
The NPCs to unlock realms are located at the following:
Unlock Sandydale Basin
Location: Horatio inside the Hermit Hut at 22, 13
Realm: Knotwood Valley
Location: Xarix at 90, 82
Unlock Barren Downs
Location: Verolia inside the Travellers Monument at 96, 91
Location: Undead Pirate at 79, 82
Realm: Barren Downs
Unlock Three Glaciers
Location: Famed Explorer at 45, 10
Location: Grug mok at 2, 62
Realm: Three Glaciers
Location: Tanas Caluse at 20, 57
Unlock Seven Sisters
Location: Explorer Vakesh at 81, 94
Unlock Royal Hollow
Location: Commander Roberts 99, 67
Realm: Seven Sisters
After a realm is unlocked use World > World to switch realms.