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).

To find the pH of a strong acid-strong base titration, you can consider that the final solution is neutral, as the reaction between the strong acid and strong base forms water (H2O). Therefore, the pH at the end point of the titration is 7, indicating a neutral solution.