Osaka has one of the most comprehensive rail networks in Japan, combining the city's own Osaka Metro (nine lines) with private railways from Hankyu, Hanshin, Kintetsu, and Nankai, plus JR services. The subway system alone covers the entire city adequately; the private lines add speed for suburban and regional destinations.
A single-day subway pass (¥620) is good value if you are making more than two or three trips. The ICOCA card (equivalent to Tokyo's Suica/Pasmo) works on all rail and bus services across the Kansai region and is the most practical option for multiple days. Taxis are metered, plentiful, and reasonably priced. Cycling is popular in the flatter parts of the city; several bike-share schemes operate, including the Docomo Cycle scheme.
For exploring the wider Kansai region, the Osaka Amazing Pass (1 or 2 days) covers unlimited subway travel and free entry to some attractions. The Kansai Thru Pass covers subway and private rail across Osaka, Kyoto, Nara, Kobe, and beyond.