The best option is to ride a train. Many of the Trans-Siberian trains pass through Nizhny Novgorod with the quickest journey being served by the Sapsan only just less than four hours from the Moscow Kursky rail terminal
Buses provide another option with the state owned bus companies offering a regular service and connections to other major cities. The Kanavinskaya bus terminal is the starting point in Moscow to travel to Nizhny, but be warned the buses travel very slowly, and it can take about 10 hours to make the journey.
The fastest way of getting around Nizhny is to ride the underground Metro, both lines travel to Moscow, one line traveling from Burevestink, the other from Park Kultury prices are usually within the 16 Ruble range.
Taxis are always an alternative, and there are several companies located in Nizhny which can be called. Trolley buses and Trams are a further option within the city to enable you to get around, but probably you will need the assistance of an English speaking Russian to help you get around the city.