Question

Classify the following as: Weak or strong acid, weak or strong base, salt (from weak/strong acid and base). If appropriate, split the formula into its component ions.

HNO3

So far, all I have is:

HNO3 + H2O -> NO3^- + H3O^+

And I know HNO3 is a strong acid. What would I do next?

You have HNO3 correct although I would not react it with water; the question states to split it so I would write
HNO3==> H^+ + NO3^-
You didn't post the other acids/bases/salts.

No, I have to classify them was weak bases/acids, salts, etc. We were told to write the appropriate expression with water. So that's why I reacted it with that. So I need to use my expression to classify them.

Also, how would I recognize a weak base?

OK if that's the instructions. You have classified HNO3 as a strong acid and that is correct. You still didn't write any other substance to classify. Read your post.

Oh wait, I was confused. I thought I had to split each thing into its forumla with water and then classify them all. Sorry about that.

I can classify the strong/weak acids and bases, but how would I know if something is a salt?

If it isn't an acid and it isn't a base, it is a salt. Weak acid and weak bases will have a Ka or a Kb listed in set of tables for them.

If that is necessary, then
HNO3 + H2O ==> H3O^+ + NO3^-
HNO3 is the acid; NO3^- is its conjugate base. H2O is the base and H3O^+ is its conjugate acid.