The Exile games were the first ones. They are quite old. They were later remade as Avernum 1-3. Then, the story continued with Avernum 4-6, which were brand new games. Now, the Avernum 1-3 are getting old, and it can be tough to run them on newer machines (especially newer Macs), so Spiderweb is remaking Avernum 1-3. So far, only the remake of Avernum 1 has been released, and it's called Avernum: Escape From The Pit.

So yes, Escapte From The Pit is a modern remake of Avernum 1, using an engine similar to (identical to?) Avernum 4-6. So you could potentially just start with Escape From The Pit, but if you do that you'll find that Avernum 2 and 3 look much older by comparison. While Avernum 1-3 are oldschool, I think they are excellent games so I'd advise you to just start with those. Then you can have Escape From The Pit to look forward to later, and each game will be slightly newer than the one before so you'll never have to go "backwards" technology-wise.

If you want to wait a while, there will eventually be modern remakes of Avernum 2 and 3 also, but they will probably not be out for a few years, as Spiderweb is currently working on Avadon 2.