Rosh Hashanah is the New Years day. It starts on the first day of the month of Tishri and ends after the second. Technically, it runs from sundown the first to sundown the third of Tishri, so it started at sundown on Sept 13th and ended at sundown the 15th. The name literally means "The head of the year" or "The first of the year". Shana Tova is "happy new year to you". There are longer and more formal phrases that are used, but Shana Tova is the common greeting for the new year.
Rosh Hashanah is also called Yom Teruah, which means "Day of making noise". It is simply called the "Day of Trumpets". Blowing the shofar in celebration is done on this day. I took an ancient Oliphant up to Ashokan with me and blew it often to celebrate.
Yom Kippur is on the seventh day of Tishri, which was last Tuesday evening thru sundown Wednesday. It is the end of the period of contemplation, atonement, and prayer following New Years. The name means "(the) Day of Atonement". It is the symbolic day that God closes the "book of life" for the next year. Those who have lead worthy lives and atoned for their past sins will be "Written in the Book". The day is the most holy day of the year, and is called "The Sabbath of Sabbaths'. It is often spent as 25 hours of prayer and fasting, while one reflects on his life, sins, and makes atonement as well as plans to change things. There is no "Happy Yom Kippur" as it is not a day of joy and celebration. The greeting or response on this day is, "May you be written in the Book".
In the period between Rosh Hashanah and Yom Kippur, all previous grievances, debts, and arguments that have not been resolved should be take care of. I call my Jewish friends and ask them, "If I have done anything in the past year to harm or offend you, I ask your forgiveness." They respond in kind, with mutual forgiveness offered by both of us. The important part, whether Christian , Muslim or Jewish is that forgiveness is final and permanent. You must let it go and make the changes needed to avoid a repeat. The forgiveness from God, Jesus, or Allah is also complete and permanent,.... as long as you accept it and make the necessary changes. The phrase, "Forgive and Forget" is not just a sound bite....it should be a way of life.