Start in the parking lot outside Observatory Train Station, on the mountain side. This is at the bottom of Trill Road in Observatory. You can see the bridge and the 'Obs Lifestyle' pop-up shop at the end of the parking lot.
There is usually enough parking for you to leave your car here. If you take the train from the Muizenberg side, simply exit the station. If you come from the Cape Town side you will have to cross the railway via one of the two subways.