How are you interconnecting the tasks? If you have three tasks one after another, and you move the start date of the first one, the others will not necessarily be rescheduled unless there are dependencies set up.
An example:
I level the resources:
Now I move the start date of the first block out a few days and level again:
The following tasks don't get moved back in time because there is not a Finish->Start dependency between the first one and the subsequent ones. That means the other tasks can be scheduled to start as soon as resources permit. Notice how Task 8 gets started but not completed, as the big first task has a higher priority in scheduling as it appears first in the list. When it finishes, then the remainder of Task 8 can be done.
If you add Finish->Start dependencies to the tasks so that Task 1 must finish before Task 2 starts, and Task 2 must finish before Task 3 starts, then level again, you get this:
and moving the start date of the first task will automatically move the whole schedule that depends ultimately on that first task. You want to set as few dates as possible, because they won't be automatically updated as the schedule changes. If you set the start date on some task to 1 week from today because that's when you "know" that a resource will be finished with the previous task, and then it turns out that estimate was wrong and the resource is available days earlier, you'll have to adjust the start date(s) on that resource's tasks or have it sit idle. The program can do the most for you if you express the constraints on the scheduling with dependencies rather than hard-coding all sorts of dates (which invariably need to be changed!)