I didn't explore Blackreach much during the Elder Knowledge quest, and I also didn't complete the A Return To Your Roots quest. It took quite a while to fight through Alftand to get there the first time so I'm hoping there is a quicker way to get in?
Initially, you can only enter via one of the "Great Lifts": Alftand, Mzinchaleft, or Raldbthar, and only with the Attunement Sphere from Septimus Signus. As you adventure below Blackreach, though, you will discover 4 additional entrances via elevators back up to the surface. These supplemental entrances are initially barred with a gate, and therefore inaccessible before you've started spelunking.
As per the UESP, here are the potential entrances into Blackreach:
- Great Lift at Alftand - (Elder Knowledge suggests you enter through here)
- Great Lift at Mzinchaleft
- Great Lift at Raldbthar
- Alftand - (outside the rest of the ruins)
- Mzinchaleft - (outside the rest of the ruins)
- Raldbthar - (outside the rest of the ruins)
- Tower of Mzark - Map Marker not visible by default, shows up after you exit with the attuned cube for Discerning the Transmudae
While several of these locations are fast-travelable-to, the Tower of Mzark is likely to be the quickest way into Blackreach, as it's associated "wing" is quite small, with no enemies.
By contrast, the Great Lifts, like Alftand, or Mzinchaleft, still have goodly portions of dwemer dungeon ruin between their elevator and blackreach proper, which will replenish their enemies (and loot) over time. Since Mzark doesn't have enemies to begin with, it will always be enemy free.
I know this answer comes quite late, but the other answers are wrong.
Alftand isn't the only way in. You can access Blackreach via Alftand, Mzinchaleft or Raldbthar. At the bottom of each Dwemer ruin is the same access mechanism that lowers the stairs to a door that leads to Blackreach, all you need is the Attunement Sphere from Septimus Signus. The quest "Transcending the Transmundane" places a quest marker at Alftand, but you may choose to ignore that and descend another way.
In my experience, Alftand is the longest path down, and Raldbthar is the shortest. However, Alftand is definitely the most rewarding because it's more interesting than the other two ruins.
After you initially access Blackreach, you can exit via the three great lifts (Alftand, Mzinchaleft and Raldbthar) that open up very close to their respective ruins. These are the fastest ways back in, but you can't go down to Blackreach through them until you come up and open the gate that only opens from the inside. There's also the Tower of Mzark lift, but it requires you to go through a short passage instead of leading straight to Blackreach.
I carry usually 10 wooden plates around because you never know when you have a dwarven ruin you need to explore the think about this is if you happen to find the barred ( caged ) way down to the ruin where your headed to and you don't wanna fight all that shit to get there if you drop the wooden plate and then pick it up by holding A ( Xbox I'm not sure how you hold items on Ps3 ) and you keep it held in front of you as you approach the gate move around against it plate might drop a few times but finally you'll find the right spot and you glitch through the doors that's how I got into blackreach and the rest of the ruins quickly