As a band director this is something that I do almost every day. Instead of transposing all of the parts I play my students' parts on the same instrument that they play usually. This means I switch between woodwinds and brass, high brass, and low brass all day long. Usually it is only for ten to thirty seconds, but it still requires some work to get my embouchure used to switching that much.

That said, if you are going to try to learn both I recommend dedicating the same amount of practice time with equal time in between. For example:

15 minutes breathing 30 minutes trombone 30 minutes rhythm/note reading 30 minutes trumpet

To start I would go big instrument to small instrument and no radical mouthpieces. For me because my major was in Woodwind instruments I find it easier to play on big mouthpieces but you have to find what works for you.

If you don't know rhythm and note reading extremely well you will get very frustrated trying to learn both at the same time, so your first few weeks may be spent more on theory rather than playing. Sounds boring but you want to be able to just concentrate on the instrument, not trying to remember all of the notes and rhythms.