Since it isn't possible to determine which database you're using to make that query happen, I'll suggest you to build your query string using prepared statements and paremeterizing your values to a PDO object based on what you may read in PHP.net documentation on the subject.
Binds a PHP variable to a corresponding named or question mark
placeholder in the SQL statement that was used to prepare the
statement. Unlike PDOStatement::bindValue(), the variable is bound as
a reference and will only be evaluated at the time that
PDOStatement::execute() is called.
As you'll see, that way you won't have to bother converting your arrays and variables to string before accessing them, plus, you grant security to your query statements.
So, instead of implodeing a string, you'll have something like this:
<?php
/* Execute a prepared statement by binding PHP variables */
$stage = $_POST['stage'];
$formats = $_POST['formats'];
$topics = $_POST['topics'];
$stmt = $db->prepare('select * from resources where stage LIKE % :stage % and formats LIKE % :formats % and topics LIKE % :topics %');
$stmt->bindParam(':stage', $stage);
$stmt->bindParam(':formats', $formats);
$stmt->bindParam(':topics', $topics);
$stmt->execute();
?>
EDIT: as you updated that you're using MySQLi, it'll be no different.
$stmt = $mysqli_db->prepare('select * from resources where stage LIKE % ? % and formats LIKE % ? % and topics LIKE % ? %');
// assuming all your params are strings
$stmt->bind_param('sss', $stage, $formats, $topics);
$stmt->execute();
As using mysqli, since it's an unbuffered sql query handler, you should store your results if you're looping simultaneous executions with $stmt->store_result();
Any doubts about how to use mysqli and pdo objects, methods and properties can be easily found in php.net documentation (linked above).
Of course, it's just a suggeston of better practices based on your apparent needs, but you can still use the implode function to achieve your string.
