Nick Childs (Euphonium Music) and James Welsh (some old Fairies CD from about 10 years ago) have both done excellent recordings of Curnow's Rhapsody for Euphonium. Although I highly doubt you'll be able to find a free legal recording of it. In that case it might be best to ask an experienced euphonium player in a lesson.
But as a rough guide (off the top of my head, haven't looked at it for a few months now), it opens with a cadenza, with the piano joining around bar 8. The whole thing needs to be lyrical and well phrased. The importance is the contrast you need to strike between the opening and the march section (where the tempo changes to 120+).
Think of Leonard Falcone, whom it was written for. I was told that as a player (I'm yet to experience his playing myself yet) he used a lot of vib in slow melodies, and really hit the accents when they came around in march sections. It is possible the piece was written to highlight these strengths. Although I wouldn't advise over-doing it in either case!