1. Apart from " Happy New Year" do people greet each other other ways?
It depends. If it's January 1, "Happy New Year" is a traditional greeting. If it's at a party on the evening of December 31, you can say things like "Cheers!" or "Here's to 200X (whatever the new year is)!"
If it's any other time of year, there are a variety of greetings people use. Hello!, Nice to see you., How are you?, What's up?, How are things?, etc. It depends upon how formal the occasion is and how well you know the person you're greeting.
2. At a wedding what to say to the bride and the groom?
The standard response would be "Congratulations!" You could also add "I'm so happy for you two" or "I wish you both the best of luck." If you're a friend of the bride, you might add something like "Your dress is beautiful" or mention how moving the ceremony was. (As a rule, the happy couple has spent a lot of money on their wedding, so any positive comments you can make about the flowers, music, clothes or whatever will be much appreciated.)
3. When a person is ill what should I say to him?
It depends upon how ill the person actually is. If he has something from which he will recover (measles, a broken leg, bronchitis, etc), "Get well soon" or "I hope you feel better very soon" are traditional statements. If he is seriously ill and there is no chance for a happy outcome, it's hard to know what to say. "Is there anything I can do for you?" or simply "I'm thinking of you" are some ideas.
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