Deep Space, also called Dark Space, is the G4+ reach of Federation, Romulan, Klingon and Independent territory. You need Operations 38 to enter. The whole region is under Fog of War: you cannot see inside a system until you have a ship or base there. The only way in is the main storyline. Clear the chain of golden-bordered missions (Buying Silence, Distant Memories, Folding, Decipher, Warzones, Syndicated Science, Perpetual Motion and several more) that ends with The Primeval. There is no shortcut.
The Delta Quadrant sits at the top of the Galaxy Map. It came with the Voyager Arc and brings new systems, hostiles, mining and missions, including your first encounters with Species 8472. It has two entry points. Border of the Alpha Quadrant (Ops 26+) has planetary missions, Hirogen Hostiles, Lost Hostiles and Species 8472 Formation Armadas. Depths of the Delta Quadrant (Ops 34+) is where you meet USS Voyager for the core story missions, against tougher hostiles. Some systems stay hidden until you complete a specific mission, such as Waste Management in the Think Tank system. A blank patch of map may just be locked.
The Veil is G7 space (Ops 71+) that players from several servers in your region share. You can preview it on the Galaxy Map. You can only enter it by ship from a G7 system. Station-to-ship transfer does not work, and your station can never relocate there. In the Veil you can see, interact with and fight players from other servers. Players from your own server look different, so you can tell them apart. G7 daily goals are in the Veil, and it is where you earn the resources to start a Surge for your alliance.