I'm trying to figure out how to start reading the Avengers comics, and have found some pretty good suggestions - however, I'm a little confused about how all of the story arcs fit together. Do the events of the Ultimate Avengers play into other arcs such as New Avengers or Avengers Disassembled at all? Or are some of the arcs in completely different universes and so have no connection with each other? There are several points that seem to be where most people recommend starting: New Avengers, Avengers, and Avengers Disassembled. I've also seen some recommendations for picking a character and just kind of going from there. It's all a little confusing...do not all of the storylines connect with each other?
The Ultimate Avengers is part of Ultimate Universe which is designated as Earth-1610. This is a completely separate continuity from the bulk of Marvel Comics which exist in a separate primary reality/continuity (Earth 616).
As such, the Ultimate Avengers do not fit in with the other titles you mentioned, as those others exist within the Earth-616 continuity.
The Ultimate continuity was created as a reboot of the primary continuity to do away with all the back history and baggage that comes with it, so that newer readers could more easily pick up and start reading without feeling like they had missed anything. You can identify the books in the Ultimate continuity from the "Ultimate" in their titles. (There may be a few limited series that do not use the "Ultimate" designation in the title, but I can't think of any.)
As a new reader, the Ultimate universe is a decent place to start, but may find yourself wanting to switch over to the main continuity for a more robust experience.
Related question: What are the main differences between the numerous Avengers comics