There are a lot of buses run along Botany Road, travelling to the city, Eastern Suburbs and South through Mascot. There are stops near the corner of Collins Street, the intersection of Epsom Road, Johnson Stree and another near the corner of Beaconsfield Street. Taxi's frequently travel along Botany Road and O'Riordan St travelling to and from the airport. Closest train station is Green Square at the junction of O'Riordan St and Botany Rd
Green Square station about 5 minutes' walk away - trains come every 15-20 min off-peak (and more often during peak hour) and it is one stop away from Central station. There are also buses running along Botany Rd (e.g., 309 310 to city, 370 to Coogee, 348 to Bondi, etc - check Google Maps). Hope that helps!