What are the early theatrical troupes of Elizabethan England compared to today? The early actors of Shakespeare’s times would regularly do what?

In Elizabethean England, actors formed companies.  For
example, Shakespeare was a member of The Lord Chamberlain's Men which was later renamed
The King's Men when they got royal patronage from James
I.

Shakespeare was a shareholder which meant that he owned
a share of the company.  These share holding actors were like mastercraftmen.  When a
shareholding actor left, he would sell his share to another master actor.  For example
when Will Kemp left, he sold his share to Robert Armin.  Since each of these comic
actors were different, the role of the fool changed in Shakespeare from the buffoon to
the thoughtful fool.  When Shakespeare retired, he sold his share to John Fletcher, a
playwright.

Other parts were taken by journeymen actors. 
It could be argued that the apprentices were the young boys who played the young female
roles since it was not thought to be fitting for females to be on stage.  The theatres
were located across the Thames in an area of drinking, gambling, bear & bull
baiting, houses of ill repute, etc.  Not a fit place for a
lady.

Like actors today, these actors would be trained to
use their voices.  Physically they would need to learn to move effectively.  They would
learn how to dance and use various weapons.  An actor whether then or now must be
phyically fit.

Since the plays were written in blank verse,
it was easy for them to memorize.  They would need to keep somewhere between 30-40 plays
in their head since a play could be revived upon request and there were no brush-up
rehearsals.

There was no director as we have today.  Stage
directions in Shakespeare are contained in the language itself.  Since these actors were
master craftsmen, they understood how this worked.  If they had a question, Will was
right there to ask.  The stage directions we find in todays Shakespeare were added by
editors.  Shakespeare wrote only a few.  The most famous is "Exit Antigonus followed by
a bear" in The  Winter's
Tale.

Compared to today, there are companies
like the Royal Shakespeare Company in Stratford-upon-Avon where a company of actors work
with a director to explore how to perform Shakespeare's plays.  However they do not own
shares in the company but are contracted for a season.  This is also true of the
current company of actors at Shakespeare's Globe in
London.

Actors then and now basicaslly do the same things. 
They train and take classes.  They rehearse and go home and learn their lines.  They
study.