How can I join multiple SQL tables using the IDs?

You want something more like this:

SELECT TableA.*, TableB.*, TableC.*, TableD.*
FROM TableA
    JOIN TableB
        ON TableB.aID = TableA.aID
    JOIN TableC
        ON TableC.cID = TableB.cID
    JOIN TableD
        ON TableD.dID = TableA.dID
WHERE DATE(TableC.date)=date(now()) 

In your example, you are not actually including TableD. All you have to do is perform another join just like you have done before.

A note: you will notice that I removed many of your parentheses, as they really are not necessary in most of the cases you had them, and only add confusion when trying to read the code. Proper nesting is the best way to make your code readable and separated out.

SELECT 
    a.nameA, /* TableA.nameA */
    d.nameD /* TableD.nameD */
FROM TableA a 
    INNER JOIN TableB b on b.aID = a.aID 
    INNER JOIN TableC c on c.cID = b.cID 
    INNER JOIN TableD d on d.dID = a.dID 
WHERE DATE(c.`date`) = CURDATE()

You have not joined TableD, merely selected the TableD FIELD (dID) from one of the tables.