If by first note you are talking about a MIDI note and wish to keep its relative time position with the following notes, I would go in the Event/List Editor to do the following:
-Select the first note of the region.
-Use the Shift-F key command to select (also) all following events in that region.
-Use your mouse to scroll down the time position value of the first note down
to 1 1 1 1
Proceeding that way would move all the events in that region (notes, etc...) back in time, keeping their relative positions intact from one another.
Consider that that procedure would disalign the synchronisation of the other regions content (if you have other regions already recorded) with the one you just edited. To avoid such, take note of the amount of time you pulled back the notes you edited and apply the same (amount of) time shifting to the other (already recorded) regions.
To ease that time correction of these other (eventual) regions, proceed from the Event/List Editor top level. There you should see a list of the regions in your project (instead of the notes events). The procedure to apply the time shift correction is similar to the one described above (for the notes) but applied to the regions. However, before applying the time shifting correction to the selected regions, make sure to deselect the region containing the first note you had performed the starting correction; otherwise you will defeat the whole process.
One additional important detail, make sure that no MIDI events (notes, regions, etc) is starting before the 1 1 1 1 time position. Logic will behave erratically otherwise. If you happen to have something before the 1 1 1 1 time position in your project, fix that by moving everything later in time, proceeding with the above described method. That "no event before 1 1 1 1 time position" issue in Logic is applying in all situations (not only in this very case).
All the above could seem quite tedious and complicated, but once you get the gist of it, and did it a couple of times, you will find the trick pretty useful and straigtforward especially if you use key commands...
