Should I think in terms of concert pitch playing the Bb Trumpet? Like if I play C,E,G should I actually think Bb,D,F? I'm having trouble seeing the advantage of thinking a note is something it is not, but all trumpet music is written a whole step up... I think... I 'm still a beginner, so I haven't looked at that much trumpet music yet.
When I started playing trumpet, I soon realised that everything I played was a tone lower than written, which was fine if the other parts were written accordingly. however, stuff I wanted to play that was in a certain key wouldn't work, as I was a tone lower. so, I learned to play in two ways: one from the dots as writ, and two as from transposing it all a tone higher, so I could then play from the same sheet as the rest of the guys who played in the band (at concert pitch, like guitar and piano). Not too difficult, and worth it.
You should start there while you're learning the basics, thinking of Bb as C, as others have said, but you probably shouldn't stop there. The disadvantage of not thinking in concert pitch is fairly small as long as you only play in wind bands and brass bands, but as soon as you step outside that, into orchestral, jazz, or nearly any other kind of music, it's going to be a real problem.
Orchestral players are routinely faced with parts for trumpet in C, but also A, D, Eb, E, and F. When the conductor is talking about the tuning on the first big D major chord at the beginning of the Sea Symphony, (s)he is not going to give it a subsidiary name for the benefit of the trumpets (and since in my score the trumpets are in F, what should (s)he call it? E major, or A major?). You could argue the clarinets would like it, but in my score two of them are in A, one in Eb, and one in Bb.
Even in a brass quintet, you've got instruments in Bb, F, C, and (depending on the choice of tuba), possibly Eb. The only way we can rehearse with any efficiency is for each player to know, and compensate for, their own transposition.
In the brass band and wind band I conduct, I ruthlessly use concert pitch for rehearsal purposes, because otherwise my musicians will never be able to play in anything other than wind and brass ensembles. In my arrangements for them, the scores (though not the parts) are in concert pitch, because I have 20 parts to worry about, and you each have only the one.
What schema you use internally is up to you, as long as you play the right notes in the right places. I don't care if you name them after famous flowers, just so long as your daffodil sharp is in tune even in peony minor. But when talking to other musicians, being able to talk in concert pitch is never a disadvantage, and often very helpful indeed.
The main reason to not think in concert pitch, and the reason we write transposed parts for wind instruments is that in written pitch the fingerings remain the same for all instruments in the family. If you know the fingerings in written pitch, you automatically can play the piccolo trumpet (It's in Eb AFAIR), the C trumpet, the alto horn in Eb. If you think in concert pitch, you need to be fluent in three sets of fingerings, apart from having to adjust to a different mouthpiece.
Even if you don't have a transposed part, transposing it is less of a time investment in a long run than learning different fingerings.
Also, the majority of wind players think and talk in written pitch so you'd have a problem communicating with them.
I also used to find the written pitch issue confusing at first but I learnt to think in it quickly. Don't worry, you'll get used to it.
I played Bb clarinet for several years, which is a similar experience. I'll talk about how I learned.
When I first started, it helped just to map the note I saw (or a sequence of notes -- clarinets have alternate fingerings to make some sequences less awkward) to a fingering. There was no thinking, "this is written C but is actually Bb, so I'll play 'C which is really Bb.'" Instead, I saw a note that happened to be written C, and I produced the fingering that made the note. It was a visual process for me, like learning how to read text silently.
Later on, I found it useful to learn how to transpose, e.g., from concert pitch to Bb pitch. When I do that, I read the concert-pitch note, and change it in my head to the corresponding Bb-pitch note. With practice, you'll start doing this for whole sequences of notes, rather than a note at a time. This is easier to do with tonal music, where you can translate a scale fragment or an arpeggio at a time.
Choir helped me learn how to read different clefs, which is a similar skill. You could try changing a sight-reading exercise to tenor clef, then practicing it. I also play mandolin and have been learning the Bach cello suites (transcribed for violin, but not much different). The 5th suite uses an alternate tuning, where the E string (on a violin; A string on a cello) is tuned down to D. The score just changes all the notes so they have the fingerings you need to play. For example, you read E F G and play those fingerings, but you hear D Eb F. It's weird when you first see it, but now it seems completely natural. In a way, it's a lot like playing a Bb instrument; you just map what you see to a fingering, and it works.
Not as a beginner, no. You should learn to react to a written note (in Bb) by fingering the transposed note. Being able to transpose in conversation (so you can understand that when the band leader refers to the Ab chord it's Bb for you) is a very useful intermediate skill, and being able to transpose fast enough to sight-read music in concert pitch (or even for an Eb instrument) is an an advanced skill that is generally not that important. But at the beginning you shouldn't worry abut transposition, beyond being aware that you're playing a transposing instrument.
As a trumpeter for 50 years, I initially learned to think in Bb, but as I started to play a variety of trumpets (in fact now I major in soprano Eb cornet) I learned to think in concert pitch, as well as Eb, D, F, etc. It was a challenge, and can still sometimes be so, but it develops over time and becomes easier as we become more experienced.