Let me first say that I'm an amateur historian with a bias. I'm a Freemason, and so I study much of the history of these societies with that biased viewpoint. That said, I feel I have some useful answers for you...

I'm referring to the Illuminati and Bavarian Illuminati (I have no clue if its one and the same)

The term comes from the Bavarian society founded by Johann Adam Weishaupt (usually just "Adam Weishaupt") and modeled on Freemasonry (as many social and philosophical societies were and still are). Weishaupt joined Freemasonry after founding his group in order to try to recruit from its ranks. What was unique about his vision was that he wanted to promote a radical and one could argue fundamentalist version of the enlightenment ideals. His ideas were later deemed to be seditious and within 20 years the group folded under the weight of its own internal strife and political pressure from the Bavarian government.

The term later came to refer to that nebulous and radical version of enlightenment philosophy, not to any particular organization. Many Freemasons, including George Washington, have cautioned against this "Illuminati" philosophy.

In the modern day, however, it has come to refer, in general and non-specific ways, to conspiracy theories regarding "The Powers That Be," ranging from one-world-government efforts to supposed shadowy organizations that control vast industrial segments.

I know our founding fathers were freemasons and in secret societies like George Washington, Benjamin Franklin and others.

Some of the US founding fathers were Freemasons. Some were not. It would be reasonable to say that Freemasonry's philosophical principles had an influence on the US, but that's hard to pick apart from the influence that the ongoing Western conversation about enlightenment political philosophy had on both of them (e.g. the works of Locke). Freemasonry developed a recognizable structure in 1717, but it didn't really solidify into the form that we know today until the early-to-mid 19th century. Before that there were a number of major changes and upheavals that make it difficult to generalize about its influences on other efforts or theirs on it.

In Europe it had great adepts such as: Beethoven, Cagliostro, Comte de St. Germain, Goethe, Mesmer and Mozart. Great musicians and influential people in that time are linked to freemasonry or secret societies like the Illuminati, what did they do back then?

The same thing they do today: maintained the initiatic tradition. This tradition goes back to the dawn of Western society, and exists in hundreds of forms. Initiation is a phenomenon that one cannot escape when looking at the history of Western culture, and its influence is felt in every walk of life.

I've heard they had meetings in secret to escape persecution from the RCC (Roman Catholic Church)

There were a large number of groups that saw Freemasonry as a threat. The Church was one, but other than in isolated areas (e.g. Spain) Freemasons was mostly left alone by the Church unless they were Catholics, in which case they were often excommunicated.

But is that why the meetings are secret? In part, yes, but in part no. [Note: this part of the answer is not historical, and is based on my personal views.] Initiation is fundamentally a very personal experience, and part of that experience is learning that it cannot be communicated to others without experiencing it for themselves. Secrecy isn't really a restriction. It's a practical reaction to the realities of the initiatic experience.

So if those same people I mentioned grew up in todays time instead of the 1700s what equivalent to the secret societies would they join if any?

[Again, opinion... I don't know that it's possible to answer this question in an objective way.] I would suggest that anyone who joined Freemasonry in the 18th century would be just as comfortable in it today, and would thrive just as much. There are some differences, but the heart of the Fraternity is still there.