Military pilot here. Takeoff speed is usually tracked via the airspeed indicator. Rotation and takeoff speeds are calculated based on weight, temperature, elevation, wind, and a few other factors. Based on those calculations, the pilot then monitors the airspeed indicator (think of it as a speedometer) and performs the takeoff maneuvers at the right speeds.
I can’t speak for commercial operations, but as a Navy pilot, we always computed both our acceleration vs speed distance, and the abort speed distance. A commonly used metric for the A-7E was “if you do not exceed 100knots within 4000′, then abort the takeoff (runways are typically marked every thousand feet). That heuristic was very conservative, of course. If you have reached 100 knots in 4000′, you’d likely know something was wrong much earlier. However, a fully loaded takeoff on a super hot day out of Albuquerque…well that’s when you need to triple check your numbers. 😉