I used to be scared to death of sight-singing exams: I simply didn't understand how anyone without perfect pitch could 'guess' what the next note would sound like. With time I discovered my own little secret; I' don't know if it'll work for you, but now I'm always the best in my class and I ace all my exams.
First, you have to be able to identify interval skips. There is a trick to this, too: treat the first note of any interval as the tonic, and then see if the second note is part of that tonic’s unaltered major scale; if it is, then the interval is major or perfect. For example, let’s say we’re in any key, and the skip upward form E-G# comes up; what is it, a major third, or a minor third? Treat the first note of the interval (E) as the tonic, and think: Is G# within the major scale of E major? Yes; therefore it is a major third. Had the interval been a skip upward from E-G natural, the answer would have been no, G is not within the E major scale; therefore it is a minor third. It doesn’t matter what key signature you are in, always treat the first note of the interval as the tonic.
Once you've got this interval thing clear, the rest is just a real cheap trick that's gonna make you laugh, but it works! Think musically: a major third upwards from any note mirrors the first two notes in the left hand of Bach's 1st C major prelude to the Well Tempered Clavier. Whenever I’m confronted with a major fourth upward, I think of the opening upbeat to Chopin’s Nocturne op 15 no. 1; a major sixth upward is the upbeat to Chopin’s Nocturne Op 9 No 2. It’s best to use only the opening of a melody that you know real well, and it’s easier if it is from a tonal composition (avoid Debussy or Schoenberg, for example).
Now you're probably thinking: how can I think of two different melodies in my head at the same time, the one I’m sight-singing and the one I’m using as interval reference? Trust me, it’s not that hard at all (unless I'm some kind of a wiz, which I seriously doubt). After a while the intervals come to you quite naturally, and this method will greatly enhance your key transposition skills. Practice half an hour every day. I mastered this method in about 4 months. I learned with a book called ‘Ear Training and Sight Singing’ by Maurice Lieberman, which gives you thousands of little melodies to practice with; but sometimes I just play any note on the piano, and try to sing any given interval above or below; that helps also.