It would dissociate into a triethylamine cation and chloride anion.
Bromide, but it is just a simple, obvious mistake.
wouldn't salts of triethylamine dissociate differently depending on the ion paired with triethylamine?
No (there are some fine prints, but they can be safely ignored here).
For instance, would I use the same Ka (that of triethylamine) for triethylammonium chloride and triethylammonium bromide?
Yes.
It may happen that the counterion itself is a weak/acid base and its reactions have to be taken into account as well (think ammonium acetate, or something like NH
4HCO
3, where HCO
3- is both an acid and base at the same time). It doesn't change the fact that each ion has its own Ka/Kb which doesn't depend on the identity of other ions - quite the opposite, it uses exactly this approach.
Maybe I don't have a clear understanding of how all of this works in general... what should I review to get a better grasp on this? You mentioned the Bronsted-Lowry theory, but is there anything else that would be helpful?
GenChem 101 - equilibrium and acid-base equilibrium.