You start them at the same time and end the semiquaver after a semiquaver duration and the quaver after a quaver duration.
Details depend on the instrument in question.
In the last bar shown you would play all the notes as quavers. It's written as it is in order to show there are two parts. If the part was split between two instruments the top part would have quavers, the lower part semiquavers (or maybe quavers). But for a single part the notation used is easier.