A car is essential in Arlington. The city has no public transport system of any significance — it is, by land area and population, the largest American city without a public transit network. Everything you want to reach is spread across a wide suburban grid that is impossible to navigate on foot.
The entertainment district — AT&T Stadium, Globe Life Field, Six Flags Over Texas, Texas Live — is compact enough to walk between once you are there. Parking lots surround all the venues and are plentiful if not free. On major game days (Cowboys, Rangers playoffs, big concerts), leave extra time: the approach roads to the stadium area back up significantly. Rideshare is the easiest option for stadium events.