For many reasons, Shakespeare, except for the history plays and The Merry Wives of Windsor, did not set his plays in England. The story of lovers kept apart by feuding families goes back to the story of Pyrmus and Thisbe in Greek mythology.
Shakespeare wrote this play in a protestant England. Since Friar Laurence is a vital character to the action, it was necessary to set the play in a catholic country.
Verona during the Renaissace was the perfect time and place for Shakespeare to expose the dangers of hatred and violence which could easily be found in his own world. The War of the Roses was a recent memory. What is the difference between the Lancasters and the Yorks and the Montagues and the Capulets? By placing the play in a foreign country, he could cover himself against criticism.
A hot summer's day in England is nothing like a hot summer's day in Italy. The climate itself plays a part. Dramatically, this makes better sense.
The story mainly takes place in Verona, Italy, which is a real location in northern Italy today. In the story, though, Romeo is banished from Verona by the prince as punishment for slaying Tybalt and goes to live in Mantua, Italy in exile.
As for the time period, Shakespeare wrote the play around 1595, so it's possible that the story takes place at that time as well, but the play never directly states the exact year. Others have suggested that it could date back to the 14th or 15th century. It could even be older since it's believed Shakespeare adapted earlier versions of this story into his own play.
The setting contributes to the story because Verona is the location of the long-standing feud between the two fighting families, the Montagues and the Capulets. It is also Juliet's home, and this is why Romeo gets so upset when he is banished-he actually says death would be better than being banished from Verona - in other words, not allowed to see his love.
The story takes place in Verona, Italy. It is a relatively small city, but one that is dominated by the feud between the Capulets and the Montagues.
The story of Romeo and Juliet itself grew from a Greek play and then several other works with similar themes, and the fact that it takes place when it does allows for the realistic existence of two great families that exist in some ways above the law of the city. It also took place in a time when the city-state was a pretty common form of government so the Prince had the power to do things like banish a subject like Romeo and enforce it.
Don't forget, when considering setting in Romeo and Juliet, that Shakespeare was writing a play to be performed live on stage. This means that the setting can really only be the time and place in which the audience experiences the production. Romeo and Juliet is a script intended for performance, not a novel frozen in time and place.
The other answers to your question are correct technically, but think about it theatrically. Do you know the play/movie West Side Story? This isn't exactly Shakespeare's text, but it stages the story of R&J in New York City in the 1960's among rival gangs. Because Shakespeare's script is so amazing (and very popular), it has been staged with all sorts of settings, some of the most recent highlighting the strained relations between those of Muslim and Christian or Jewish faiths. The feud is a powerful element in R&J, and still exists as a powerful force in our society today.
The truth is, this play could be set right in your own neighborhood or on your school's campus. What if you considered the possibility of the setting being Verona High School? How might this bring you closer to understanding the characters and their choices?