Why do we use hydrochloric in the test for sulfate ions but nitric acid for halide ions?
The link above which I used for reference mentions that for the test
  for sulfate anions, hydrochloric acid (HCl) is used to ensure that
  there is no presence of carbonate ions. For the test for halide ions
  (F−, Cl−, Br−, I−, etc), nitric acid (HNO3) is used instead, for the
  same purpose.


Good question, but this is not a universal practice. In reality, the choice of acid depends on the actual sample. Qualitative tests, like these, have become merely academic exercises due the to advances in instrumental analysis these days. As you said, one cannot add HCl to destroy carbonates in a sample which is to be tested for halide ions. Dilute nitric is the only choice, because you would not use sulfuric acid in the case of halides because (i) your test sample may contain ions which will precipitate with sulfate (Ba, Ag, Pb, Ca etc.). Concentrated nitric acid would be a poor choice as well because it will oxidize iodide to free iodine.

An excellent book, which is far better than any online resource, is Vogel's Qualitative Inorganic Analysis. It should be available in any good library and it will give an in depth view of qualitative analysis.