Put POST values into cookies.

You might want to put them into the session array for short-term storage, or store them in a database for longer-term storage.  There is no design pattern that puts meaningful data into a cookie.  Cookies are like database keys -- they are pointers to the information, not the information itself.

Here's an article about the PHP session.  HTH, ~Ray
https://www.experts-exchange.com/articles/11909/PHP-Sessions-Simpler-Than-You-May-Think.html

Firstly I would put all those values in the form into an array - say data
data[select_time1]
data[select_time2]
...

Now you can do the following. The code below will check both POST and SESSION for a data element. The first one found is used - if none is found an empty array is used. This is then merged with the default values so the result is a filled array that has the correct values

<?php
session_start();
$default = array(
  'select_time1' => '',
  'select_time6' => '',
  'protiens' > '',
  'mon' => '',
  'fri' => ''
);

$data = loadData($default);
print_r($data);
/*
  This function gets the data from the post - if there is none
  it gets the data from the Session - if there is none it 
  defaults to an empty array.

   It then loops through supplied default array and merges the
   values from the default with those it already has from POST
   or Session. It returns an array that is guaranteed to have values
   for all the required values - obtained either from POST or  Session
   in that order
*/
function loadData($default)
{
  $data = !empty($_GET['data']) ? $_GET['data'] : false;
  if ($data === false) {
     $data= !empty($_SESSION['data']) ? $_SESSION['data'] : array();
  }
  
  foreach($default as $key => $value) {
    if (!isset($data[$key])) {
       $data[$key] = $value;
    }
  }
  $_SESSION['data'] = $data;

  return $data;
}

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If I'm not misunderstanding the question, the values are already in an array -- it's $_POST that contains the request variables.

Maybe something along the lines of this design would be helpful.  It seems to handle both form validation and remembering the good values, and on my Chrome browser, it handles the "back" button sensibly.  Haven't tested it on other browsers.
https://iconoun.com/demo/form_highlight_errors.php

<?php // demo/form_highlight_errors.php
/**
 * Demonstrate how to highlight errors in form input, and remember valid inputs
 * Client is asked to put in a value
 * If the value fails our test we show an error message
 * We put a marker next to the input control on the form
 * We turn the form border red
 * SEE http://www.w3schools.com/CSS/pr_class_visibility.asp
 */
error_reporting(E_ALL);


// THESE STYLE ELEMENTS ARE SET FOR THE SCRIPT INITIALIZATION
$error_abc = $error_xyz = $error_any = 'hidden';
$boxer_abc = $boxer_xyz              = 'black';


// CAPTURE AND NORMALIZE THE POST VARIABLES - ADD YOUR OWN SANITY CHECKS HERE
$abc = isset($_POST["abc"]) ? trim(strtoupper($_POST["abc"])) : NULL;
$xyz = isset($_POST["xyz"]) ? trim(strtoupper($_POST["xyz"])) : NULL;


// IF ANYTHING WAS POSTED, VALIDATE IT
if (!empty($_POST))
{
    // VALIDATE THE 'abc' FIELD
    if ($abc != 'ABC')
    {
        // VALIDATION FAILED
        $error_abc = $error_any = 'visible';
        $boxer_abc = 'red';

        // BECAUSE THIS FAILED VALIDATION, REMOVE IT FROM THE FORM
        $abc       = NULL;
    }

    // VALIDATE THE 'xyz' FIELD
    if ($xyz != 'XYZ')
    {
        // VALIDATION FAILED
        $error_xyz = $error_any = 'visible';
        $boxer_xyz = 'red';

        // BECAUSE THIS FAILED VALIDATION, REMOVE IT FROM THE FORM
        $xyz       = NULL;
    }

    // DO WE HAVE INPUT FREE FROM ANY ERRORS?
    if ($error_any != 'visible')
    {
        echo "CONGRATULATIONS - ALL VALIDATION PASSED";
        die();
    }

    // OTHERWISE... OOPS - WE HAVE ERRORS AND MUST SHOW THE FORM AGAIN
}

// IF NOTHING WAS POSTED, OR IF THERE ARE ERRORS, WE NEED NEW CLIENT INPUT
$form = <<<ENDFORM
<style type="text/css" media="all">
.error_any { visibility:$error_any; }
.error_abc { visibility:$error_abc; }
.error_xyz { visibility:$error_xyz; }
</style>
<pre>
<form method="post">
<span class="error_any">PLEASE CORRECT THE FOLLOWING ERRORS</span>
<span class="error_abc">YOU MUST ENTER 'abc' IN THIS FIELD</span>
PLEASE ENTER "ABC" HERE: <input style="border-color:$boxer_abc;" name="abc" value="$abc" />
<br>
<span class="error_xyz">YOU MUST ENTER 'xyz' IN THIS FIELD</span>
PLEASE ENTER "XYZ" HERE: <input style="border-color:$boxer_xyz;" name="xyz" value="$xyz" />
<br>
<input type="submit" />
</form>
ENDFORM;

// WRITE THE FORM WITH THE APPROPRIATE CSS STYLES ON THE ERROR MESSAGE FIELDS
echo $form;

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it's $_POST that contains the request variables.

Yes but you have to test and access each one at a time. If you use a form array you can get them in one go.

How about this?

<?php
session_start();
$_SESSION['POST'] = $_POST;

Select allOpen in new window

Personal preference - I bring all form data (that needs to be validated, saved to db) in an array - I know that everything I need should be in there and I don't have to unset anything. I can then set other POST variables for control / routing etc that don't interfere with the form data. Allows for quick extraction of a subset of $_POST without having to go through the POST array - as I said personal preference - I use it extensively and it works for me in the frameworks I use.

The variables all contain the values I need. I just have a trouble time switching through the page because when I click on a button it takes me back, therefore the session is gone because I'm back to the first page. This is why I want to store my sessions into a cookie so they won't go, when I switch through the pages.

Also I have a form that I allow users to keep submitting, how can I append the form values into the session as well?
For example:

$protien when I first submit the form i chose chicken. Then later I want to chose another protien from a different meal so I click lamb. how can I make the protien get the lamb but still have chicken inside it?

like:
$protien = 'chicken'
$protien - 'chicken', 'lamb' //after i post form again

therefore the session is gone because I'm back to the first page

If this is really happening to you, there is something else going wrong in your application.  Please read the article about how sessions work.  It is required knowledge for PHP developers.  The session, once established, should remain in place as long as you're still on the same web site.  If it's losing data when you "go back" you almost certainly have a design flaw or logic error in your app.

Sessions allow you to store complex types (Arrays and Objects)

Sometimes, but not always.  Please see the cautionary note here:
http://php.net/manual/en/ref.session.php#120110

Yes stdClass is a special case - and for other objects you need to include the Class file so that PHP knows how to interpret the data.

The point is that the serializing into and out of the session is handled by PHP whereas with a Cookie - you have to do it yourself.

Yeah, PHP has not got all of this right yet.  Instead of serializing, JSON-encoding may be a better choice sometimes.  Wish this were a magic answer, but it's PHP, and so it's not ever going to be a magic answer. Anyway, you've got an answer and some links to the essential learning resources, so I'll sign off now.  Good luck with your project!