If you cycle this card, the cycling ability goes on the stack, then the triggered ability goes on the stack on top of it. The triggered ability will resolve before you draw a card from the cycling ability.

The cycling ability and the triggered ability are separate. If the triggered ability doesn't resolve (due to being countered with Stifle, for example, or if all its targets have become illegal), the cycling ability will still resolve and you'll draw a card.

You can cycle this card even if there are no targets for the triggered ability. That's because the cycling ability itself has no targets.

Grixis Sojourner's triggered ability may target Grixis Sojourners itself. By the time the ability triggers, Grixis Sojourner will already be in your graveyard.

The triggered ability acts as a cycle-triggered ability or as a leaves-the-battlefield ability, as appropriate. The player who controls the triggered ability is the player who cycled the Sojourner, or the player who last controlled the Sojourner on the battlefield.