Any tips on how to make it sick, so to speak, when trying to internalize the distance between notes?
There are three ways you can easily get those intervals in your head.
Singing the intervals will make learning them much more easier and effective. Try this before doing your interval exercises:
- Pick one interval you are having troubles with.
- Play the interval in the instrument you are most familiar with (or a piano app or whatever)
- Sing the interval in two different ways: "la, la" (or whatever your favorite syllable is) and using the interval name: "la, la, minor second".
- Repeat until you become more familiar with it. You don't have to internalize it in one sit, you can do it just a few times a day and you'll eventually get it in both your mind and your voice.
You should be singing the intervals doing the actual exercises too:
- Listen to the interval given by the app.
- Sing it.
- Tell the direction of the interval (ascending, descending, at the same time) and the interval itself.
Singing is not only very useful when internalizing intervals, but for developing your musicality in general. Interval recognition exercises that include singing are a basic part of most improvisation programs, where being able to sing a specific interval or a given string of intervals is very useful, to give one example.
Associate an interval with a part of a song that you know
If you are having troubles memorizing a specific interval, try finding it in a song you like, and make a mental note about it. It'll stick like glue.
For example: the major third interval can be found in the first notes played by the bass in Radiohead's "National Anthem". Once I noticed it and made a mental note about it, it was very easy to recognize the interval in isolation.
The perfect fifth can be found in the first two notes of Top Gun's main theme, and that's what first came to my mind when I listened to it in isolation (I don't even remember the movie that well), and that's how it stayed in my memory from the start. It also works with Jurassic Park's main theme.
Once you associate an interval with a part of a song that contains the interval, a song that you like or can otherwise easily remember, memorizing intervals will become many times easier.
In your major 2nd and major 3rd case in particular, you can think about the major scale. Can you sing it? If you can then the 2nd major interval is the first two notes of the major scale, you already know it. If the interval sounds like it is starting a major scale from the root, then there's your major 2nd (it could also be starting other scales, but the major is by far the most common to know and the one most people can sing).
Twenty minutes a day, every day, should be the minimum practice time and frequency if you are looking for serious progress. You don't have to practice intervals only, you can practice scale and chord recognition too.
Are there some intervals I should focus on first?
Find two intervals that are easy to recognize for you, and start with those. I started with perfect 4th and perfect 5th, then added the tritone, and kept adding intervals every time I felt like I could handle one more.
If you can't recognize intervals that are close together yet, try with other pair of intervals that are farther apart. In your case, try major 2nd and perfect fourth instead. The difference should be more evident. Once you can distinguish that pair, try adding other interval to the mix.