В POST отбражается только sumbit. толи лыжи не едут,толи...
<form id="msform" action="submit.php" method="POST">
<!-- progressbar -->
<ul id="progressbar">
<li class="active">Personal Details</li>
<li>Adress Details</li>
<li>Comment</li>
</ul>
<!-- fieldsets -->
<fieldset>
<h2 class="fs-title">Personal Details</h2>
<h3 class="fs-subtitle">Tell us about you</h3>
<input type="text" id="first_name" placeholder="Your First Name" required="" />
<input type="text" id="second_name" placeholder="Your Second Name" required="" />
<input type="text" id="phone" placeholder="Phone" required="" />
<button type="button" name="next" class="next action-button">NEXT</button>
</fieldset>
<fieldset>
<h2 class="fs-title">Adress Details</h2>
<h3 class="fs-subtitle">Enter Your Address</h3>
<input type="text" id="street" placeholder="Street" required="" />
<input type="text" id="building" placeholder="Building №" required="" />
<input type="text" id="city" placeholder="City" required="" />
<button type="button" name="previous" class="previous action-button-previous">PREVIOUS</button>
<button type="button" name="next" class="next action-button">NEXT</button>
</fieldset>
<fieldset>
<h2 class="fs-title">Comment</h2>
<h3 class="fs-subtitle">Fill in your comment</h3>
<textarea id="comment" placeholder="Comments"></textarea>
<button type="button" name="previous" class="previous action-button-previous">PREVIOUS</button>
<button type="submit" name="submit" class="submit action-button">SUMBIT</button>
</fieldset>
</form>
У тебя в input'ах нет параметра name. POST отправляется по name, а не по id. Например:
<input type="text" name="street" id="street" placeholder="Street" required="" />
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