Arrange the following acid in increasing order of strengths #(HCO_3 )^- , (H_3 O )^+ , (HSO_4) ^ - , HSO_3 F#?

We cannot consider the hydronium ion, #H_3O^+#, as it is an hypothetical entity, whereas the the other species are stuff you could put in a bottle and characterize. We conceive that when an acid is put into water, a strong acid protonates water to form the hydronium ion, #H_3O^+#.

I am breaking my own rule, and not looking at metrics, which I should properly do, but here we can use induction.

The strength of an acid relates to how far the following equilibrium lies to the right:

#HA(aq) + H_2O(l) rightleftharpoons H_3O^+ + A^-#

For stronger acids, the equilibrium lies further to the right. Again this reaction illustrates my reluctance to consider the acidity of #H_3O^+#, the acidium ion in water, as it is the result of acid-base behavior in water.

We have a choice between anions and neutral species, and, reasonably, the anions, bicarbonate, and bisulfate should be weaker acids than the neutral species.

So the strongest acid is fluorosulfonic acid, #HSO_3F#, then bisulfate, #HSO_4^-#, and then bicarbonate, #HCO_3^-#. I urge you to look up #pK_a# values for these acids.