Brass instruments are great. I hope you have lots of fun learning how to play, and that they open doors for you.
To try to answer the question, the way the mouthpiece behaves when it's not inserted into the instrument is very different to the way it behaves when it is inserted.
When it is inserted, the instrument imposes "slots" on the pitches you can produce, which means it drags you to a resonant peak, for instance if you have no valves pressed you can most easily play C, G, C, E, G, Bb C... Unless you're really determined, you won't be able to play a good note in the gaps between those slots.
But there are no such slots when the mouthpiece is out of the instrument. You can swoop between pitches no problem. It's a nice exercise to go from as low as you possibly can to as high as you can, without the sound breaking up in between.
When the mouthpiece is inserted, don't fight the instrument. If you've set the tuning slide to Bb tuning then when you play your lowest open, non-pedal note, it's a concert Bb. Try to play right through the middle of the note, so you're not bending it flat or sharp; not fighting against the slot.
Sheet music for transposing instruments is helpfully transposed for you. The composer knows what concert pitches are required, and writes on the sheet music the notes that you have to perform on your instrument to produce those concert pitches.
So on a Bb instrument, you'd perform a written C (first ledger line below the stave in treble clef notation) and the sounding note would be a concert Bb.
Usually you're thinking about performing the notation, so you're thinking that you're playing a written C, and not thinking that you're performing a concert Bb
But over time as your experience grows you can mentally switch between thinking about the notated pitches and concert pitches, and for brass players it's a valuable skill to be able to perform from music notated in concert pitch without first having to write it out into transposed notation.