Which is started first depends on several things, including the configuration of the airplane (what pumps and accessories are on what engine, for example), and the operation involved (two engine taxi vs. four engine taxi, etc).
Generally I always started 1, 2, 3, and 4 in that order. In aircraft with inboard engines having hydraulic pumps, then inboards first, or alternating inboards first.
When taxiing on two engines, usually inboards only, and with short taxi's, then all four engines turning to warm up in the short taxi distance available.