The Mishna offers us important advice, especially for Rosh Hashana, when it says (tractate Avos 2:4), "Don't separate yourself from the public." When a person prays as a part of the tzibur, the public community, he or she gets judged together with the entire public. Such a trial is much easier, for it's less exacting on the individual. Therefore, it's advisable to pray in an upright community with upright people, preferably led by a righteous person. For this reason, many people travel great distances to pray with a tzaddik on Rosh Hashana. May all those travelling to pray by Rebbe Nachman's gravesite in Uman, Ukraine have a safe trip.
In Israel, there's no need to worry if you don't belong to a synagogue; there a tens of places where one can pray with a tzaddik, have a place to stay with festive meals, and it won't cost you a cent. Miron is a fantastic place to spend Rosh Hashana. In Ashdod alone, a guest can pick from any number of inspiring places; for yeshivishe davening, you can go to Grodno. For chassidishe davening with a Rebbe, you can go to Pittsburg, Neshchiz, Chernobel, or Melitz (the Belzers and Gerrers all go to Jerusalem, while the Vishnitzers go to Bnai Brak). If you like Sfardi davening, you can pray with the Baba Sali's grandsons, Rebbe Raphael or Rebbe Yekutiel Abu Chatzira (each has his own shul) right here in Ashdod.
If you're in America, and you either don't have the money for High Holiday tickets or you don't know where to go, or maybe you're an embarrassed newcomer to traditional Judaism, there's no need to worry. My very special brothers at Chabad sent me this Rosh Hashana services link, where you can find the shul nearest you that will welcome you with open arms, including a place to eat and a place to stay.
Praying with the public is important, but sometimes it's worthwhile just to close your eyes, bury your face in a scarf or hankie (if you're a woman) or in your tallis (if you're a man), and pour your heart out to Hashem in your very own words. This way, you have the advantage of the public aspects of prayer (public merits, shofar, kaddish, kedusha, etc.) with the emotion and intensity of your deepest meditation. Don't hold the tears back.
An old story attributed to the Baal Shem Tov zatza"l tells of a orphaned country bumpkin who couldn't read the aleph-bet, yet believed in Hashem with simple and pure faith. He came into town to join in the Rosh Hashana services, but didn't know how to pray a word. When others cried out in prayer, he simply looked up at the heavens, and said, "Dear G-d, I can't read or pray, but I know it's a holy day. I don't know what the others are saying, and I don't even know how to express myself. I do know how to imitate a rooster, so I dedicate this call to you...Cockadoodle-doo! Cockadoodle-doo!" As the orphan crowed, the tears streamed down his cheeks, for this was his personal prayer from the inner walls of a pure heart. The Baal Shem Tov said that the orphan's prayers pierced the heavens, and caused all harsh verdicts for the coming year to be rescinded.
Hashem never turns away anyone who comes to Him with a with sincerity. Rebbe Menachem Mendel of Kotsk said that nothing is more whole than a broken heart. May you have a sweet New Year 5775, and may all your prayers be answered for the very best, amen.