The reason why I ask this question is because I'm wanting to get a tenor trombone with a F trigger and I would like to know if its the same notes as a traditional trombone, because I play the traditional trombone.
A typical slide trombone is pitched in Bb. This simply mean that when the slide is in 1st position, the harmonic series is based off of a Bb.
For a student in early to mid grade school, the vast majority of the time this will be a tenor trombone.
A tenor trombone can also have what is called an F attachment, or quart-valve. This lowers the pitch by a perfect fourth to F when the valve is engaged. It is often referred to as Bb/F. Meaning when not engaged, the first position partials are based of Bb. When the valve is engaged, first position becomes the F overtone series.
Effectively this means that you will be able to play Bb to C (second line and second space in bass clef) in 1st position -> trigger 1st position. This can be abbreviated as 1 -> T1. The bottom space F is also playable in first position. This is because the extra tubing is like adding 6 positions worth of length.
Because the horn is longer when the F attachment is engaged, the positions are longer. You can play chromatically from Bb 2nd line of the bass clef down to C, two lines below the bass clef with the following positions:
Note that because the positions need to be longer, T3 is closer to 4, and by the time you get all the way out to T6, the distance to be in tune would be the same as 7th position when the F attachment is not engaged.
So the short answer to your question is a Bb/F trombone is "the same" as a "traditional trombone" when the attachment is not engaged. However, the attachment adds some extra notes and techniques not available on a horn without an attachment.
Yes, you'll play the same parts, the same notes will come out. You'll gain a little agility in the low register, and have access to a few notes between low E and pedal Bb that no sensible composer writes for tenor trombone anyway (and which the bass trombone can make a better job of)! You'll also have a heavier instrument to carry, and a slightly less open sound, even when not using the valve (though this isn't a big issue on a good instrument). The F attachment is an asset, but maybe not SUCH a great one as you imagine.