Presumably Four to Six Times
We know that students take the Hogwarts Express at the beginning of the year and at the end of the year, as these journeys are documented in most of the books. We also know that students can go home for the Christmas holidays, and Harry in fact does leave Hogwarts for Christmas in both Order of the Phoenix and Half-Blood Prince. In Order of the Phoenix he takes a portkey to Grimmauld Place, and takes the Knight Bus back to Hogwarts:
"You have all used a Portkey before?" asked Dumbledore, and they nodded, each reaching out to touch some part of the blackened kettle. "Good. On the count of three then... one... two..."
They were to return to Hogwarts on the Knight Bus the following day, escorted once again by Tonks and Lupin, both of whom were eating breakfast in the kitchen when Harry, Ron, and Hermione arrived there next morning.
In Half-Blood Prince we are not told how they got to the Burrow, but they return to Hogwarts via the Floo Network:
As he and Ron had left for the Burrow early the next day, he had barely had time to wish her a happy Christmas and to tell her that he had some very important news when they got back from the holidays.
Late in the afternoon, a few days after New Year, Harry, Ron, and Ginny lined up beside the kitchen fire to return to Hogwarts. The Ministry had arranged this one-off connection to the Floo Network to return students quickly and safely to the school.
However, it would appear that these travel arrangements were an exception rather than the norm. In Order of the Phoenix they had to leave Hogwarts under emergency circumstances and wanted security on the way back; in Half-Blood Prince the Ministry apparently set up a Floo connection for security reasons as well. Indeed, earlier in the series we do find the Hogwarts Express being used for the Christmas holidays.
In Chapter Twelve of Chamber of Secrets we find the following statement:
There was almost a stampede to book seats on the Hogwarts Express so that students could go home for Christmas.
In Deathly Hallows when the Ministry was already under Voldemort's control (so there would be no need for extra security traveling for Christmas) we again find the train used for Christmas:
"The only people in real danger are the ones whose friends and relatives on the outside are giving trouble. They get taken hostage. Old Xeno Lovegood was getting a bit
too outspoken in The Quibbler, so they dragged Luna off the train on the way back for Christmas."
Hogwarts also has Easter holidays. In all the years Harry attended Hogwarts, though, I don't believe we ever see anyone going home for Easter. However, in Prisoner of Azkaban a conversation between Fudge and Harry implies that students could go home for Easter:
"Ah, you're worrying about the reaction of your aunt and uncle?" said Fudge. "Well, I won't deny that they are extremely angry, Harry, but they are prepared to take you back next summer as long as you stay at Hogwarts for the Christmas and Easter holidays."
Harry unstuck his throat.
"I always stay at Hogwarts for the Christmas and Easter holidays," he said, "and I don't ever want to go back to Privet Drive."
And in Deathly Hallows, though Harry no longer attends Hogwarts, we do hear about Malfoy and Ginny being home for the Easter holidays:
"Follow me," said Narcissa, leading the way across the hall. "My son, Draco, is home for his Easter holidays. If that is Harry Potter, he will know."
"... lucky that Ginny’s on holiday. If she’d been at Hogwarts they could have taken her before we reached her. Now we know she’s safe too."
"We lost Luna at Christmas, and Ginny never came back after Easter, and the three of us were sort of the leaders.
However, we are not told how Ginny or Malfoy returned home. We might assume that the Hogwarts Express is used for Easter just like it is used for Christmas, but on the other hand it is possible that if so few people tend to go home for Easter then they wouldn't bother running the train.