It depends. The first beat of a bar is always down (called the “downbeat”). In almost every meter, the last beat is up.

Before a piece starts, there is usually at least one preparatory gesture. This is called the “upbeat” when the piece starts with a complete measure. But it is not the first beat of the first measure, only preparatory. There are times, though, when a piece starts with an incomplete measure and, then, the preparatory gesture is the beat that would have gone before whatever is the first beat. (That is, if the piece begins on the second beat, the preparatory is the first or downbeat. If the piece begins on the third beat of a measure, then the preparatory gesture is whatever the “theoretical” second beat would have been.) Furthermore, note, in very fast tempos, you usually give at least a full measure of preparation.

Finally, remember that up and down are not static. That is, you never come to a full stop. When the baton (or hand) goes down, it bounces a bit in the direction of the next beat. Some conductors conduct in a very small space or, even, a horizontal line. I tend to make gestures that clearly look like the four points on the compass (in a 4/4 measure).

Here are two beat patterns typical of me: