I'm a euphonium player, but for jazz band at school I play trombone. Everyone in jazz band is required to play an improv piece only this is my first year doing trombone, so I'm not so certain. Should I just ask to do it on the euph or should I try it, and if so, how can I make it work?
You are asking, (1) which instrument will you use, and (2) what advice can we give you for getting started with soloing, which you find rather daunting at this point.
You can buy a book and play along CD. Once you've learned a tune well, you're reading to try improvising with both instruments while listening to the play-along track. Hopefully this will make it easier for you to choose your instrument to use in this ensemble.
Download some drumlines (from marching bands), then play along, improvising. This is a gentle way to get started, since there aren't any chords to worry about playing notes from.
Basically, what you're asking is how to play a solo on an instrument that you are technically not very skilled at (yet) and still make it work.
What you need to do is
Play fewer notes. Longer notes and more space between them (e.g. play 3 notes in 2 bars, then rest for 1 bar, etc.)
Keep it simple. Choose 1 or 2 notes on each chord which you want to hear (e.g. a nice tension).
Practice playing the guide tones. Then arpeggios. In the beginning, for practice, fill the bars with them so you know how to play them. Then, pick only small parts of them and leave space in between those parts.
The answer to whether to play your solo on trombone or euphonium is one of personal choice. If you want to develop your improvisation skills, trombone might be the better choice since it is a much more common instrument in jazz. You probably won't get invited to many big bands as exclusively a euphonium player.
Regarding the question of how to improve your skills, my best piece of advice is to transcribe and learn great solos. Choose solos at your level of proficiency, so for instance, you might start with the first chorus in Miles Davis classic "Walkin". Simple F blues that requires very little technique.
I just made a video about transcribing where I use a JJ Johnson solo. That particular solo might be challenging for you, but I think you'll get some value from the video: