What is Ulysses about? Is it worth reading?

One of the most radical innovations of Ulysses (1922) is that in conventional narrative terms very little happens. Over a single summer's day (June 16, 1904) we share the lives of three Dubliners: Stephen Dedalus  (a recently bereaved young graduate), Leopold Bloom (a middle-aged sales representative of Jewish origin) and Molly Bloom (unfaithful wife of Leopold and occasional singer). All the action takes place in and around Dublin.
How does it start?
Stately, plump Buck Mulligan came from the stairhead, bearing a bowl of lather on which a mirror and a razor lay crossed. A yellow dressinggown, ungirdled, was sustained gently behind him on the mild morning air. He held the bowl aloft and intoned:- Introibo ad altare Dei.
Martello Tower, Dublin where Joyce lived briefly
It is early in the morning. Twenty-two year-old Stephen Dedalus awakes in the Martello Tower, where he is temporarily living with a medical student, Buck Mulligan. An Englishman, Haynes, is staying with Mulligan. 

Like Homer's Telemachus, Stephen is edgy and anxious and suspects he is being ‘usurped‘ by treacherous friends. He is particularly upset at overhearing Mulligan's unkind remark about his mother being ‘beastly dead’. Tellingly, he is less concerned about his mother's memory than  the 'insult to me'.

The end of A Portrait of a Young Man left Stephen ready to 'flee' his native city and the 'nets' of politics and religion. A year later he is less self-assured but in a state of internal exile, physically in Dublin but ready to leave Ireland behind. Again mirroring Telemachus, Stephen suspects the motives of those who try to dissuade him, telling his nationalist former teacher ‘history is a nightmare from which I am trying to awake’.

You can hear a short extract from the BBC version here - Part 1
Stephen doesn't sound much fun. But the story is about Bloom, isn't it? 
After three chapters of Stephen's morose musings ('impossible person') we meet our unassuming hero, Leopold Bloom. He is eating breakfast:  
Mr Leopold Bloom ate with relish the inner organs of beasts and fowls. He liked thick giblet soup, nutty gizzards, a stuffed roast heart, liverslices fried with crustcrumbs, fried hencods' roes. Most of all he liked grilled mutton kidneys which gave to his palate a fine tang of faintly scented urine. 
After taking breakfast-in-bed to his wife, Bloom spends his day plodding around Dublin fulfilling various humdrum obligations: the funeral of an acquaintance, sales visits, a walk on the beach.  We accompany him, entering his inner world through a 'stream of consciousness':

listen to ‘Leopold Bloom’ on Audioboo

In crucial respects Bloom is an outsider in a very insular society. Though he has converted to Catholicism in order to marry Molly, he is never allowed to forget his Jewish origins. The Citizen pointedly asks which 'nation' he claims allegiance to. Even those not hostile to him, mock his cuckold status. 

What is the connection between Bloom and Stephen? Do they know each other?

Not directly. The only connection is that  Bloom is acquainted with Stephen's father, Simon Dedalus. 

National Maternity Hospital where Stephen & Bloom meet
But Dublin is a small city and throughout the day Bloom and Stephen, circle each other, finally crossing paths at the National Maternity Hospital. The kindly Bloom is visiting Mrs Mina Purefoy (in her third day of labour as described in extract above) while Bloom is drunk after carousing with Buck Mulligan. Bloom rescues Stephen from a brawl and the two men visit Nighttown - the red light district of Dublin.

Bloom is haunted by his lost son ('if Rudy had lived') and his paternalism is touched by the drunk and vulnerable Stephen. Though the two men are divided by class, age and religion there is an unspoken filial bond between them.

Why is Ulysses considered to be a groundbreaking novel?
Joyce, in a bravura display of technical brilliance, frames the novel around Homer’s Odyssey. For Hades, for example, we have Paddy Dignam's funeral while the brutal Citizen is Cyclops.  Joyce is very playful within this structure - it is the Citizen, for example, who calls Bloom  'one eyed'. 

At the end of the novel, Ulysses (Bloom) returns home to his Penelope (Molly). His odyssey is successful in that she remains faithful to him, emotionally if not physically. In her soliloquy, a stream of consciousness without punctuation that mimics the moments between waking and sleeping, her thoughts continually return to Bloom. In this sense he triumphs over Boylan and all his rivals.

Joyce with his editor Sylvia Beach in Paris
No novel prior to Ulysses had so radically challenged the conventions in both form and subject matter. For forty years the frank sexual content caused the continual problems with censors in the UK and the US but this feels less daring to us today.

What is still astonishes is the musicality of the prose - Joyce was an accomplished tenor and he has an unparalleled ear when it comes to mimicing sound. Bloom's cat's progressively more plaintive demands for food are transcribed precisely:  "mkgnao", "mrkgnao" and "mrkrgnao".

Is Ulysses a good read. Should I take it to the beach?
Maybe not to the beach - certainly not without a Kindle! Ulysses is very long (over 265,000 words) difficult to follow in parts, almost unreadable in others. Sometimes cleverness overwhelms content - when the prose mimics the evolution of the English language, for example.

It's hard to escape the feeling that for many intellectual trophy rather than a genuinely loved friend. Though it is a 'must-have' for the educated person's bookshelf, it has also been described as the 'most unread' novel of the Twentieth Century. Many get no further than Buck Mulligans showing off in Latin in the opening paragraph - a clear signal that you are in for what is called in Hollywood 'homework'.  

So is reading Ulysses worth the effort? I would answer with the last of all those words, Molly Bloom's resounding, 'yes'. And if pushed to present evidence, I would cite Leopold Bloom perhaps the greatest creation in modern literature. 

Is Ulysses the best place to start with Joyce?
Dubliners is a more accessible entry point - some of the stories 'Ivy Day in the Committee Room' and 'The Dead' in particular  are amongst the finest in the English language. They also introduce characters and themes that reappear in Ulysses.

For a brief introduction to the autobiographical elements of Ulysses see  this New Yorker review .
Ulysses by James Joyce - Softback (US)
Ulysses: Annotated Students' Edition (Penguin)
Dubliners by James Joyce
Biography - James Joyce (Oxford Lives)