I heard people mention training to recognize intervals many many times. What I am a bit confused about is this. Do you train to recognize intervals from a given note to the root, or to the preceding note? It seems to make more sense to learn to recognize intervals from the root, but I'd like to confirm this is the case.
Normally people that go this route train to recognize intervals from the preceding note. This way they can (theoretically, at least) see any piece of music and sight-sing it by singing the consecutive intervals.
But for what it's worth, my experience is that this is not the most efficient way to learn to sing accurately. (It may be helpful for post-tonal music, but even then I'm not convinced. And I'm assuming you're more interested in singing tonal music than post-tonal.) If you really want this intervallic approach, you have to learn the intervals from every scale degree, because an ascending perfect fifth from scale-degree 1 is a very different animal from that same ascending perfect fifth from scale-degree 2, from scale-degree 3, from scale-degree ♭3, etc.
In my opinion, the better way is to learn to sing functionally. Instead of singing that weird ascending perfect fifth from scale-degree 3 to 7, just sing scale-degree 3 and then sing the leading tone. Easy as pie!
As others have mentioned, it's important to be able to identify intervals both bottom-up and top-bottom (and really any other configuration as well). If said I randomly words a in sentence, brain your probably still would sense of that make.
Intervals may be broken down into three classes:
Class 1 = Perfect (8, 5, 4, unison)
Class 2 = Consonant (3rds / 6ths)
Class 3 = Dissonant (2nd / 7ths)
This of course includes major / minor / dim / aug alterations as well. (Minor 3rd is still consonant as inverted it's a maj 6th). By establishing a hierarchy in your brain, you'll be able to more easily identify and distinguish interval types.
Most ear training for intervals consists of two notes at a time, and you simply identify how far apart the two notes are, without any regard for whatever 'root' may or may not be implied.
However, it is also useful to identify intervals sequentially, and as other answers say it is good in that case to know both what the distance is from the previous note AND what the distance is from the root of the sequence of notes, if that can be determined. (It can't always.)
Hope that helps. If not, perhaps you could edit your question to clarify what you are asking.