How do you eliminate the noise if the sound of the water running is extremely loud when the toilet bowl fills?

If the toilet is continuously running, you need to replace the flush valve assembly, or part of it, in the tank. If the flushing sound itself bothers you, you might be able to buy styrofoam insulation to line the inside of the tank. This will reduce the noise a bit while the tank is refilling, and will also reduce condensation on the outside of the tank during humid weather.

Consider also replacing the toilet with a newer low-flush type. These flush quickly, refill quickly and use considerably less water, and are often quieter in operation than old-style toilets.