I have to find the correct acid-base indicator to use for the following titrations (methyl orange with end point @ pH 4, bromthymol blue with end point @ pH 7, or phenopthanlein with end point @ pH 9).

I have several problems given to me, but first I'm trying to figure them out using an example in the book.

The book gives an example of the titration of a weak base-strong acid system with NH3 and HCl. It shows the equation as NH3 + H+ --> NH4+. Then it says that K= 1/Ka, or in this case 1/5.6 x 10^-10 = 1.8 x 10^9. The pH is given as 4.78. How was the pH found? I tried it myself as
5.6 x 10^-10 = x^2/1.00 M (the book says 1.00 M is assumed), but I end up with 4.63 as my pH. Any help would be appreciated.

Your question isn't quite complete; however, I think I know part of the problem. Also, I worked backwards to find what I think you didn't post. First, the titration of a weak base vs a strong acid gives a salt (like NH4^+) and that is what is present at the equivalence point. Therefore, the pH is determined by the salt's hydrolysis (with water).
NH4^+ + H2O==> NH3 + H3O^+
(NH3) = x
(H3O^+) = x
(NH4^+) = 0.5 (if they assumed the concentrations were 1.00 M then the concn of the salt, at the equivalence point, is just 1/2 that since an equal amount of HCl titrant was added.
Second, the Ka for NH4^+ is 5.6E-10. The K is NOT 1/Ka. Kw = KaKb and since Kb for NH3 is 1.78E-5, then Ka = 1E-14/1.78E-5 = 5.6E-10. Using the 0.5 M for the salt at the equivalence point gives me
5.6E-10 = x^2/0.5. Solving for x gives me (H^+) = 1.67E-5 and pH for that is 4.776 which rounds to 4.78. I hope this helps get you started. Please post the entire problem if you have other questions.

I have another question. I understand how to calculate the pH of strong acid-weak base and vice versa, but I'm confused when it comes to strong acid-strong base. I have the titration of hydrobromic acid and strontium hydroxide. I know that the bronsted lowry net ionic equation is:
H3O+ + OH- --> 2H2O
But that's not how I'd find the pH, is it?

yes. pH of a stong acid/strong base titration is just the pH of H2O (since neither the anion nor the cation is hydrolyzed).