Volgograd does have an international airport, but its global reach doesn’t extend much further than its name unfortunately. There are direct flights to Yerevan in Armenia, so the ‘international’ is meant in the most liberal way possibly. Direct flights to Moscow, St. Petersburg, Sochi, Krasnodar and Samara are available however. The airport is a 25 minute ride from the centre of town, although expect to pay double for any luggage you may have.
As expected, you can get to Volgograd via train from pretty much any western Russian city worth its salt. Moscow (20 hours) and St. Petersburg (34 hours), and you’ll have to change in the big cities in order to head further west. Those heading in the eastern direction can find routes to Irkutsk (92 hours), while many major Central Asian cities are also on the timetable. The station is handily located on Railway Station Square and is a Stalinist piece of work. Fans of communist architecture may well get excited about such things.
The small bus station is situated behind the train station, and that is about as much information as you’ll need. Stick to trains for getting in and out.