это простой jquery, использующий метод stepUp для добавления дней
today = new Date();
day = today.getDate()
month = today.getMonth() + 1
year = today.getFullYear();
day = day < 10 ? '0' + day : day;
month = month < 10 ? '0' + month : month;
today = new String(year + '-' + month + '-' + day)
$("#second")[0].min = today;
$('#first').on('change', function(e){
date = new Date($('#first').val());
$('#second')[0].valueAsNumber = date.setDate(date.getDate())
$("#second")[0].stepUp(7);
})
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<input type="date" id="first">
<input type="date" id="second">