It is going to be difficult to find precise data during the initial launch, however let's see what might be available:
For SpaceX launches, their videos posted in their YouTube channel display altitude and speed, as well as visualizations of their orbits and their projected ground-track across Earth's geography. Their press releases contain additional information, including approximate times and target orbits (shape, inclination, etc). The most notable, prolific, and accurate set of reconstructions posted for public consumption is available at flightclub.io (read more in reddit).
For answers here that have used details from flightclub, see here and here and here.
For the movement of spacecraft around the Earth once in orbit, you can use two line elements available from Celestrak.org or Space-Track.org. See this answer for more about Space-Track, and this site is filled with questions and answers referring to Celestrak and TLEs, and you will find several with Python scripts that allow you to calculate trajectories from the TLEs.
As some launches continue to Earth orbit and especially to deep space, JPL's Horizons will provide calculated past and future trajectories. This site is also filled with questions and answers referring to Horizons, and you will find several with Python scripts that allow you do extract and plot orbits from Horizons data.
There may be reconstructions of historical launches, such as those of the Gemini and Apollo missions, and in fact, some of them can also be found at flightclub.io as well!