The way music schools teach this is relatively brutal and annoying, with a _lot_ of repetition and testing (eg "sing a major second above this note" and "identify the interval" questions), but I am not sure any other method works. At the same time, everyone going through an ear training curriculum does pick up decent relative pitch. This can take a year or two for college music majors, so it's not exactly a casual exercise. However, I assume the major barrier to entry is not musical aptitude but willingness to put up with bullshit, because it feels like bullshit when you are doing it.
I remember many years ago in my music lessons being shocked that some people can hear multiple notes played simultaneously. I've never found much material on learning this skill.
You can practice singing the intervals. What's a fifth sound like? You should be able to sing it. Or play it on your instrument and then sing it.
If you're willing to give the app a try, I bet it could actually be a pretty solid way to learn relative as well as absolute pitch. Just manually play "Red" before you start to anchor yourself. I've noticed some improvement in my relative pitch just by practicing it with my daughter. I'd be interested to know if anyone ends up using it explicitly for that purpose.