After reading this post you'll have a MUCH better understanding of what's going on with German articles and their genders and feel much more confident about them. You'll know why we even have words like "der", "die", "das", when to use which, and what's up with articles like "dem" and "den", too! All you need to do to learn German articles now is read on. Here we go:
1) What matters most is how the word sounds

First, when confronted with a new German noun, it's important to note that it will inevitably have one of three grammatical genders: male (masculine), female (feminine), or neuter. Kind of like "he", "she", and "it" in English, except that, unlike English, German grammatical gender has nothing to do with biological sex in the vast majority of cases - so a girl is "it", a cannon is "she", and a bouquet of flowers is "he".
Now, that seems pretty random, but it isn't.
Have you ever considered that Germans, too, get confronted with new words almost daily? And we, too, usually need to infer the gender of these words. Ask any German if the unknown word "Arachibutyrophobie" (the fear of peanut butter sticking to the roof of one's mouth) is male, female, or neuter and they will ALL say it's female. You can even make up a word that literally does not exist and Germans will (usually) agree on its gender. And that's simply because of pattern recognition:
If a new word rhymes with a known word, it will almost certainly have the same gender as the known word. It's the ending that's most important here. What does a plant have in common with a bottle, a bag, and a roll of toilet paper? They all end with the letter "e" in German, and are thus almost certainly feminine.
The good news here is that you will get dramatically better at gender the more exposure you have to the German language. You don't even need to sit down and agonizingly memorize to learn German articles! When you hear "der Kleber" (the glue), and "der Rechner" (the computer), and "der Pfeffer" (the pepper) often enough, you'll figure that "der" would also go with "Toaster", "Transporter", and "Roboter". This is almost effortless.
Aside from how a word sounds, there are some (really weird!) categories of things that also have a strong association with the word's gender. Like, rivers inside Germany are typically feminine while rivers outside of Germany are masculine! Or that alcoholic drinks are almost always masculine (even "Bloody Mary").
Of course, there are some exceptions, but broadly speaking the patterns hold true. Occasionally, even Germans can't agree on the gender of a word (Das Email or die Email? Der or das Virus? Bonbon? Joghurt?) So don't worry and trust that you'll get a feel for it over time - and it's no problem to make mistakes until you get there.
2) Why is this necessary? Can't we just dump this whole system instead of having to learn German articles?

In this picture, all three creatures are masculine (two men and a dog), but in the sentence they fulfill different roles. In German we use articles to signpost what role a person or thing is taking in the sentence. What's happening is that the blue guy is giving the red dog to the yellow guy. The blue guy would be "der Mann" because he is the doer (subject) of the action (giving). The red dog would be "den Hund" because he's the direct object of the action, he's the answer to the question "WHAT is being given". And the yellow man is the indirect object, the recipient of the give action, and would here be "dem Mann".
Wait, don't leave!
Look at this english sentence: "He gives him to him." In German that sentence would be "Er gibt ihn ihm." Even in English, we make a difference between "he" and "him" and "to him" in order to signal what's happening in the sentence. The good news here is that an English-speaker's brain is already configured for this type of structuring because we can differentiate between things like "he"/"him" and "she"/"her" - which is essentially all that's happening in German, except with articles instead of pronouns.
So, again, relax about this oft-maligned aspect of German, because we already know how to do this type of thing, conceptually.
In German, having the articles be able to transform allows us to move things around in sentences, shifting emphases without the meaning getting lost. The sentence "Der Mann gibt dem Mann den Hund" means the same thing as "Dem Mann gibt der Mann den Hund."
We Germans even put articles in front of people's names because we just love being able to express their roles in our sentences: "Den Stephan habe ich lange nicht gesehen." (I haven't seen Stephan for a long time).
If you'd like to know everything (and I literally mean everything) you need about this so-called case system in German, check out my compact video grammar workshop, and let me ease you into the concepts in a way that'll finally make it click. It's easy.
3) What's the best way to actively learn the genders of German words?
Fear not, I've got you covered. If you're itching to better learn German articles, I created a (100% free) illustrated vocabulary book featuring the 1000 most frequently used nouns in German just for you, with example sentences, and, crucially, different colors for different genders.
This book was designed to prevent German learners from wasting their time learning words they don't need, and allowing you to focus on words Germans actually use most frequently every day. My recommendation would be to go over this ebook (in PDF format): You can either tackle a page a day and memorize the words, or (and I actually recommend this more), just skim through it every day and see what sticks. Remember, the idea is to become good at recognizing patterns. I assure you, after skimming over these 1000 nouns, you'll have a dramatically improved feeling for which gender goes with which word.
In this book, the masculine words are blue, the feminine ones are red, and the neuter ones are yellow. Associate "der" with the masculine ones, "die" with the female ones, and "das" with the neuter ones, consciously, and when reading them out loud, in order for your brain and your ear to become used to the pattern. You'll be amazed at how well this works. Get the book now.
I hope this was helpful and got you started on the right path to mastering German gender! Much success and if you have any questions, contact me anytime using the contact form below!