The game starts with you limping into Earth system after a quick intro slideshow. There, you run into some "do this immediately" and really shallow plot exposition.
You have next to no fuel, so while you can leave Sol system, there won't be much to do before you dead-end.
There are 4 events in the system. The space station orbiting Earth, the Spathi hiding on Pluto, the Ur-Quan security satallite, and the stuff on Luna.
The long conversations (Spathi and Space Station, after the plot) involve people who know marginally more than you do. The Ur-Quan security satallite knows much more, but disengages after sending you a message. Luna has some clues, which only make sense in context.
You don't know it, but there is a timer on the game from the first moment you start. The deadline is reasonably far away, but events will unfold, and if you don't deal with them, you and everything you care about will be destroyed.
You are sent out to gather resources (to upgrade your ship and build new ones). In pretty much every direction, there are alien races that you'll bump into that will generate more plot. In addition, you have to come home for fuel -- and when you do, your station commander will relay some events that have happened since you where last there (be they timed (the ZFP signal), or triggered by you) that they pick up on their long-range sensors and communications satellites, or people showing up at the station as you build your alliance up.
The various alien races tend to be doing things -- other alien races respond to your meddling with change in behavior. You start civil wars, you stop invasions, you start genocides, etc. Admittedly, the races "further" from your home base are less dynamic than the ones "nearby".
As time goes on, you get hints asto when the "deadline" that is approaching is, and what its nature is. You gather technologies that make travel easier, expanding the radius you can explore, and making your ship less tied to returning home. As you fly around the universe poking into things, the various races start moving around in response, and you can ignore them or interfere.
There is no "this is the time to dick around", but in order to figure out what is going on, you need to dick around. And you need allied races for their resources and technology, and sometimes for their plot coupons.
As an explicit example, saving the Pkunk is utterly an option (be it from the Yeehat or the Illwrath). At a number of times, there are plot forks (how do you get a hyperwave broadcaster -- I believe there are 2 of them in the sector?), which have an impact on what the sector will look like post victory.