What measurements helped scientists understand 'Oumuamua's location and speed? I wanted to know how it was achieved and with what accuracy if possible.
Like most asteroids and comets, 1I/'Oumuamua's trajectory was determined entirely by measuring its position in optical images over several days. The earliest data came from automated, ground-based surveys such as Pan-STARRS and the Catalina Sky Survey, then targeted follow-ups from various other asteroid observers. The object's evident extrasolar origin motivated additional observations with more sensitive instruments including the Hubble Space Telescope. 'Oumuamua did not pass near enough to the Earth for a radar observation.
The Minor Planet Center collects those observations here. In the table of orbital elements, the 0.4 arcsecond residual is the root-mean-square difference between positions computed using those elements and positions actually observed. In the observation table, the magnitude column indicates which photometric filter was used, e.g. G or R for greenish or reddish bands of visible light. Pan-STARRS filter w passes a wide band including most of the visible band and some near infrared (Tonry et al. 2012 table 4).
JPL's estimate of 'Oumuamua's orbital elements uses the same observations and includes uncertainties. They estimate about 1000 km error in perihelion distance (q), 23 seconds error in perihelion time (tp), and 1 arcsecond error in inclination (i) and ascending node longitude (node). If you follow the Ephemeris link from there, and specify
Ephemeris Type: VECTORS
Table Settings: quantities code=2x
then HORIZONS gives Cartesian position and velocity uncertainties based on dynamical simulation. These are larger now than when it was observed, and still larger a few months before discovery.