What obstacles does Romeo face during the course of Shakespeare's Romeo and Juliet?

Romeo has a varying degree of success in overcoming the obstacles he faces in Shakespeare's Romeo and Juliet. The ultimate obstacle is to be with Juliet despite the bitter feud. Unless you believe the two are together in heaven, Romeo fails in clearing this hurdle.

The first obstacle is his unrequited love for Rosaline. He becomes depressed because Rosaline avoids his affections and decides to stay celibate. Romeo says, in Act I, Scene 1:

...

Romeo has a varying degree of success in overcoming the obstacles he faces in Shakespeare's Romeo and Juliet. The ultimate obstacle is to be with Juliet despite the bitter feud. Unless you believe the two are together in heaven, Romeo fails in clearing this hurdle.

The first obstacle is his unrequited love for Rosaline. He becomes depressed because Rosaline avoids his affections and decides to stay celibate. Romeo says, in Act I, Scene 1:

She’ll not be hit
With Cupid’s arrow. She hath Dian’s wit,
And, in strong proof of chastity well armed,
From love’s weak childish bow she lives uncharmed.
She will not stay the siege of loving terms,
Nor bide th’ encounter of assailing eyes,
Nor ope her lap to saint-seducing gold.
O, she is rich in beauty, only poor
That, when she dies, with beauty dies her store.

Romeo is much more successful in his love for Juliet. The girl is swept off her feet and they kiss within a few minutes of meeting each other. They part ways at the end of Capulet's party but Romeo is not satisfied. He pursues Juliet into the orchard and overcomes both the wall and his fear of being discovered by one of the Capulet men, who have no love for him or his name. He strengthens his love for Juliet by asking her to marry him. She agrees.

Romeo's next obstacle is convincing Friar Lawrence to marry him to the daughter of Capulet. The Friar is skeptical because just the previous day Romeo had professed his love for Rosaline. The Friar comments in Act II, Scene 3:  

Young men’s love then lies
Not truly in their hearts, but in their eyes.

The Friar, however, agrees to Romeo's request and decides to perform the wedding ceremony because he believes it may be the best way to finally end the feud between the Montagues and Capulets. The Frair says,

In one respect I’ll thy assistant be,
For this alliance may so happy prove
To turn your households’ rancor to pure love.

Because Tybalt had seen Romeo at the party he chooses to challenge Romeo. In an example of dramatic irony Romeo tries to avoid fighting Tybalt by telling him that he really loves him, but Tybalt does not understand the situation. Romeo says, in Act III, Scene 1:

Tybalt, the reason that I have to love thee
Doth much excuse the appertaining rage
To such a greeting. Villain am I none.
Therefore farewell. I see thou knowest me not.

Unfortunately, Romeo is not able to handle this obstacle and winds up fighting and killing Tybalt after the death of Mercutio. Romeo cannot help but give in to the urge for revenge after his best friend is killed. The killing of Tybalt, of course, is the beginning of the end for Romeo.

He faces three more obstacles. One is summoning enough courage to actually leave Verona and Juliet after consummating the marriage in Juliet's bedroom in Act III, Scene 5. Juliet suggests he should stay because she hears the nightingale singing outside her window signifying it is still night. Romeo wants to stay but eventually gets away and assures Juliet they will be together again one day despite her fears:

I doubt it not; and all these woes shall serve
For sweet discourses in our times to come.

After hearing the news from Balthasar that Juliet is dead Romeo quickly decides his only recourse is to commit suicide. He is able to procure poison even though it is against the law to sell. He finds a poor apothecary and bribes him into selling a dram of poison:

Come hither, man. I see that thou art poor.
[He offers money.]
Hold, there is forty ducats. Let me have
A dram of poison, such soon-speeding gear
As will disperse itself through all the veins,
That the life-weary taker may fall dead...

His final obstacle is to actually take the poison and extinguish his own life. But, after seeing his beloved dead in the Capulet's tomb he shows his resolution and drinks the deadly mixture. He succeeds conquering this last obstacle even though the reader wishes otherwise.