What are good resources for self-learning modern molecular biology concepts?
Sure it's feasible! Of course, there are lots of papers that I can't read even with a recent education, but you should be able to make a good dent by:


Getting a good modern textbook like Molecular Biology of the Cell by Alberts et al. This is becoming a standard text and gives pretty good coverage of the field. I find that it doesn't always get as detailed as I would like, but such is the nature of textbooks I think. Another textbook that some people like is Molecular Cell Biology by Lodish et al, although I don't have experience with it myself.
Watch online lectures. MIT OpenCourseWare has a pretty good molecular biology course. It may get basic for you at times, but the lecturers are well-respected molecular biologists and geneticists.
Read reviews in things like Nature Reviews to get a 30,000-foot view. Review authors often touch on concepts that primary literature assume you know. Armed with a good text, you should be able to get a good idea of what people are talking about. On a similar note, the "News and Views" commentary articles that accompany "high-impact" papers are great for giving people unfamiliar with the field a taste of why a certain paper deserves attention.
Come back here and ask!