The Economics of Interstellar Trade
One obvious possibility is trade in information and services since different planets will have different cultures and might be best at different things. (ObSF: The Dorsai series.) It might start out as free (gratis), but the people downloading movies from the Movie Planet might prefer action-adventure movies to artsy stuff and the people downloading programs from the Software Planet might prefer understandable user interfaces. Eventually, they will get around to paying for the extras.
Trade in material resources might also be possible. System X might have a high ratio of sunlight to planetary mass and System Y might have the opposite. In that case, it makes sense for them to trade atoms for photons. (The atoms can be sent by spaceship and the photons by laser.) In particular, the planets of a red dwarf might have the atoms needed to support a population in the quadrillions but might not have the energy to do so. On the other hand, Sirius might have the energy to support an enormous population but might not have any planets.
Interstellar colonization might be set off in reaction to a sunlight monopoly in the Solar System. A colony around Alpha Centauri can match the amount Big Sunlight can sell and a colony around Sirius can exceed it by a very large margin. On the other hand, Big Sunlight might try maintaining the monopoly by shooting at colonization ships.
Of course, Sirius might also become a monopoly. Most of the starlight emitted within 5 parsecs of the Sun comes from Sirius. If Sirius exploits its monopoly, there will be attempts to bypass it by starting colonies near Vega or Arcturus.
I think we have a plot as well as a background …
The best part is that this revives the Golden Age SF cliche of putting colonies near first-magnitude stars.