How to attain a deep conceptual grasp of key atomic/molecular/genetic/ notions?
I'm assuming you have a reasonable high school education otherwise you probably wouldn't have posted your question. 

Do a basic science course and/or read - maths, physics, chemistry, biology texts. 1st year university level should be good.

If there is anything you don't understand in these texts, go to more basic books - plus Wikipedia is good for the sciences (I wouldn´t like to read them on, say, the Arab-Israeli question), but I've never seen any egregious errors in the scientific area!

Then read Klug and Cummins "Concepts of Genetics" from cover to cover. 

If there is anything you don't understand in Klug and Cummins, go back to basic texts. 

I had a good background in the  sciences and then majored in Genetics with minors in Biochem, Microbiology and Immunology. I read that book from cover to cover taking notes in 3rd year (of a 4 year degree programme) and have done very well in the field. 

The book covers the history of the discoveries in a very readable and approachable fashion. Obviously it is no substitute for doing a degree in the subject, but if you really wish to at least have a good understanding of the science, you could do worse than read that text!

You could also read popular science books aimed at the "interested and educated layperson" - Stephen Jay Gould, Richard Dawkins for example. Try National Geographic and even Scientific American or New Scientist. You, at least intially, will have to make a real effort, but it is very rewarding - there is nothing more fascinating than biology generally and molecular biology/genetics in particular! Best of luck on your journey!