We generally started the left engine first.
The whole normal start/preflight check sequence was long, involving tests of the air inlet control system, wingsweep and flap interlocks and flight controls.
But to start the engines was pretty straightforward: power connected, starter air connected, electric lights test, fire warning system test. Engine start switch to the engine being started (this opened a valve on the air turbine starter and began turning the engine). As the RPM advanced above 18% RPM (minimum, we generally went for the maximum that the starter air source would give us), move the throttle for that engine from cut-off to idle, watch for fuel flow (about 400#/hour flow to start), within a few seconds of fuel flow, the TIT (Turbine Inlet Temperature) should rise. If TIT was rising rapidly (hot start), or if there was no rise (wet start), or the RPM did not advance normally (hung start), then move the throttle back to cutoff and allow the starter to windmill the engine until cool/clear.
The F-110 motor was similar, except that there was a digital gauge for RPM, with an illuminated pixel for fuel-on, so you weren't counting RPM as closely. The F-110 had EGT as a gauge, not TIT, and the limits were different. The digital engine controls on that engine made it more reliable on start than the hydro pneumatic TF-30 engine controls.
Once the left engine was running, you would crank the right, engage the bi-directional hydraulic pump, and stop cranking the right. As the engine wound down, observe that the BiDi picked up the right engine hydraulic system pressure, then cut the BiDi off to make sure that the selector switch for it worked*.
Then start the right engine as the left.
*There were two hydraulic systems, the left engine powered the combined system, which included landing gear, brakes, refueling probe, AICS, wingsweep and flight controls, and the right engine powered the flight hydraulic system, which powered AICS, wingsweep and flight controls. This BiDi check was important, it allowed either hydraulic system to power the other side, at a slightly reduced pressure. A failed Bidi could leave you without critical components in the event of a hydraulic failure. A BiDi that was engaged with a failed system would overheat on the side with no fluid. So, you had to check that it would 1. work and 2. could be shut off.
After the engines were running, you then went through a test of the AICS, which was called OBC, and the ramps in the inlet would move full deflection and back up to a stowed position for takeoff (they start moving in flight as a function of Mach number, and became relevant above 0.9 IMN). After OBC, then the wings come out to 20 degrees, landing flaps selected, flight controls checked, spoiler operation checked, flaps up, maneuver flaps selected, wings manually selected to aft, observe that they stop at 50 degrees, retract maneuver flaps, sweep to 55 degrees, then sweep to 68 and stow to oversweep for taxi. (On the ship, it was abbreviated to simply: wings to 20 degrees, landing flaps selected, quick spoiler check, then hook up into catapult, mil/AB power selected, flight controls check, and launch.)