On the top navigation pane for every Vessel and Fleet, you are given a Run Now button in the top-right. Clicking this button will cause the Vessel or Fleet to schedule a voyage immediately. You will then be re-directed to the newly created log.
For Vessel logs, live data and output will be shown without you needing to refresh.
For Fleet logs, you'll need to manually refresh the page to see updates.
Vessels may not run instantaneously, as there is still time associated with entering a job queue, spinning up servers, and installing any packages provided in Requirements.