1) If you want to move several tasks to all start on the *same* date, then you can select all the tasks you want to change (by dragging or shift-clicking) and then use the inspector to change all their start dates to the same thing with just a couple more clicks.
2) If you want to move several tasks the same number of days into the future (for example, Monday's task to Wednesday, Tuesday's task to Thursday, etc.), then it depends on how you've set their start dates in the first place.
If you select a task and it says "On Specified Date" in the task inspector, then you are going to have to change that task's date directly. This occurs when you manually set task's start dates in the inspector or drag them in the Gantt chart.
If the inspector says "As soon as constraints allow", then changing whatever the constraint is (a group start date, a pre-requisite task's end date, etc.) should have a ripple effect and move everything together.
If the inspector says "By resource availability", then you need to tweak the resource's schedule and/or relevel the project to see the desired changes.
If that doesn't help, maybe you can give more specifics or a screenshot here or via email to our ninjas:
omniplan@omnigroup.com
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Note to everyone: You should only be scheduling a task "On specified date" if there's sort of a formal, external date involved, and you don't want this task moving around just because others are. For example, a company-wide meeting, a pre-announced ship date, the day a government official is coming for an inspection, etc.