The triplets do start on a beat. All of them. And you are right that there are 12 of them in a measure of 4/4. However, make sure your math is correct. A triplet is:
ONE - 2 - 3 - TWO - 2 - 3 - THREE - 2 - 3 - FOUR - 2 - 3.
Each of the capital letters represents a beat of four. I suggest you do not try counting with "1-e-+-a". Instead, simply count "one-two-three". I usually would count the beat with an emphasis as above. "ONE-two-three-TWO-two-three" etc.
If this is difficult for you, you should be able to find a metronome that can help you get a firmer grasp of the grid. Or, you can set a metronome to a FAST pace, accent every 3 beats. Then consider every three clicks to be one beat.
If this is still difficult, you're probably just not hearing enough use of the triplet in music you know. Try listening to music with lots of triplets. Classical music has plenty of examples. In modern music, the eighth-note triplet is probably used most often in a blues context.
Keep practicing. What was frustrating and disappointing one day is amazingly easy the next, and I think learning beat divisions are a perfect example. Once you've learned it, you won't forget it.
One more thing: check to see if the technique you are using fits your goals. It's perfectly natural that you should use three fingers to play a triplet. Try with one hand, and then both hands playing the same triplet in unison. In addition, chord arpeggios going up and down can be a great way to learn the triplet because they tend to fall into a pattern of 6, which can be thought of as two triplets: