To show our anticipation and desire of Shavuot, we begin “counting the Omer” for seven weeks, until the day finally arrives where we can commemorate the giving of the Ruach haKodesh. (Explanation can be found in this article.) So what we see here is, at Passover, the Israelites were freed from their lives of slavery in Egypt; and 50 days later on Shavuot they accepted YHWH's Torah which made them a nation committed to serving God.
Exodus 34:22 tells us: "Observe the festival of Shavuot with the first-gathered produce of the wheat harvest, and the festival of ingathering at the turn of the year...."
As a side note about when to start the Omer count, which is to start on "First Fruits": There is a huge contention about which day is to be called "First Fruits" when the wave offering made and the 50-day count of the Omer begins. The contention boils down to the meaning of the word "Shabbat" in Leviticus 23:10-11. Every year, people argue about whether the word "Shabbat" means the "weekly 7th day" or the High Holy Day of the Feast of Unleavended Bread. (Many don’t realize that ALL High Holy Days are "Shabbats" in the meaning of the set-apart day of rest, no work, or convocation. See Leviticus 23:24 and 23:28 for example). To clear up the confusion, we have written this article.