The first and most important idea to note is that Kundera does not necessarily view lightness as a positive. Indeed a major point of the novel is to explore the question: "The only certainty is: The lightness/weight opposition is the most mysterious, most ambiguous of all." There is a reason the title refers to lightness as "unbearable," but this doesn't mean he comes out clearly against lightness, either. It depends on person and on the circumstances. Lightness represents our choices and our freedom, but weight represents our commitments and habits, and thereby provides meaning. Though it is not a perfect map, it reminds me a bit of some of the thoughts David Brooks has expressed recently on happiness and suffering and happiness vs. meaning.
Some people, when faced with an oppositional ambiguity, yearn to select one, which is in this case a meta-preference toward weight. Others refuse to select one - though choosing not to commit is itself a commitment. But it is not necessary to choose one or the other in all things. We can choose lightness in some areas, weight in others. We can leave our professional options open while marrying and having children, or remain single while changing career frequently. My general interest in the topic of "lightness" is in understanding how to navigate this ambiguity.
Just to get into a particular point, one notion in the book that caught me by surprise is the connection of public persona and weight. There is an exploration of whether "living in truth" means being transparent - the same in public and in private - or having a distinct private world where one need not assess the public implications of actions. Some might view the weight in the lies or at least inauthenticity of separating public and private; others would see it in having to maintain a chosen public face (whether it is chosen by oneself or others). It seems like where one feels this weight might depend on whether one is an extrovert or introvert. It is clear to me: I could not bear to have an "authentic" public persona. It would mean either having to manage the implications of being my authentic self in public, or changing who I am to control those implications. I like having a private world that I do not have to justify. This is a subjective assessment - everyone is different. But it is surely a good lens for understanding oneself.