The short answer is that Tom escaped from the prison of a Southern-woman dominated household, to an adventurous world (Merchant Marines a possibility hinted at). For the real Tom, it was a life of playwriting.
Tennessee (Thomas) Williams was obsessed with the destructive Southern culture, which includes a strong mother figure, whose influence did not extend to business but only to social standing and reputation. The “old school” manners and etiquette were blind to the human spirit, constricting in their rigidity, and entirely incompatible with an artistic, imaginative temperament. And desertion from the family is the greatest social sin; Tom and Amanda’s father, a telephone lineman “fell in love with long distance.” The implication is that Amanda’s rigid, domineering personality, formed in large part by her failed youth, has sucked all the individuality out of the family, and made a prison of their lives. Amanda is living in the past and has trapped Laura in it with her. As for Laura, there is a real question whether she will ever be free of her mother’s influence, because she is sickly and because her only advocate, Tom, has left to join the Merchant Marines.( The music from the dance hall across the alley is also a symbol of where he might escape to.) Her world is now devoid of "magic" and imagination (symbolized by the unicorn's horn breaking off, leaving as ordinary horse.)
AMANDA: I took that horrible novel back to the library- yes ! That hideous book by that insane Mr. Lawrence. [Tom laughs wildly.] I cannot control the output of diseased minds or people who cater to them - [Tom laughs still more wildly.] BUT I WON'T ALLOW SUCH FILTH BROUGHT INTO MY HOUSE ! NO, no, no, no, no !
In the play, Tom’s need for adventure is best summed up in his love of movies. When asked why, he says “I like adventure.” And in one of the most heart-rending lines in the play, when asked why he goes to so many movies, he responds “I like a lot of adventure.” It is more than coincidental that Williams himself, something of a misfit in Southern society, could so vividly depict the prison of a Southern household dominated by a mother whose own dreams were never realized. As for how complete is his escape, the final lines of the play suggest that he is not comfortable with having left Laura trapped.
Finally you ask of Amanda's "escape." She does not.