This question is similar to Is there any way to create a timestamp in a Google Spreadsheet? except here you want to date of the first edit, not the most recent one.
This cannot be achieved with spreadsheet formulas, because if a formula inserts today's date subject to some condition (nonempty cells), it will also insert tomorrow's date tomorrow, if the condition still holds.
This can be done manually, by pressing Ctrl; (Control/Command and semicolon) in Column A after making the first edit to a row.
Or this can be done automatically by a script.
Such as this one, which inserts the date of first edit to a row in the first column of that row. The date is inserted in ISO format yyyy-MM-dd. (It can be any other format supported by SimpleDateFormat.)
function onEdit(e) {
var row = e.range.getRow();
var dateCell = e.range.getSheet().getRange(row, 1);
if (!dateCell.getValue()) {
dateCell.setValue(new Date()).setNumberFormat("yyyy-MM-dd");
}
}
The script should be entered in Tools > Script Editor. It runs automatically on every edit, but it will not replace the date in column A if it's already there.
To have a script recording the date of last edit, change it
function onEdit(e) {
var row = e.range.getRow();
var dateCell = e.range.getSheet().getRange(row, 1);
dateCell.setValue(new Date()).setNumberFormat("yyyy-MM-dd");
}