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I am a beginner in music theory so please bear with me. I was looking up a document which talks about transposing music and I understand a few things I think about transposing music. But if I, for example, have a sequence of these notes:

    E D# E D# E E D# E F# G# F# G# A B A# B A# B

• how do I determine which key they are in?


• how do I transpose them to a key of D? Is this online tool accurate? https://transposr.com/
  

(It's going to be tough to explain all of this in a single answer. If you're interested in this, I strongly recommend finding a music theory text, either online or in hard copy. But I'll do my best to address it all here!)

When it comes to major and minor keys, the best way to determine tonality, in my opinion, is to determine the location of half steps. (You can also determine the location of the tritone, but really that's just a fancier way of determining the half steps.)

Major scales have a pattern of WWHWWWH, where H is a half step and W is a whole step (two half steps). Minor scales are a bit trickier, because there are three uses of minor—natural minor, harmonic minor, and melodic minor—but we can skip that for this answer.

Looking at your example, there are half steps between E and D♯, between G♯ and A, and between A♯ and B. (There's also a half step between A and A♯, but we'll address that in the next paragraph.) If we try to compare this to our WWHWWWH pattern, we see that the best fit starts on E. E to F♯ is a whole step, F♯ to G♯ is a whole step, G♯ to A is a half step, and so on.

The only trouble is the appearance of A♯. Here's where it gets tricky, but not too tricky, because we have a rule in tonal music: each note name will only appear once in a major scale. If another version of that note name appears, it will be a chromatic pitch. So since A makes sense in our E-major scale, we can view the A♯ as a chromatic pitch. All of this tells us that these pitches are likely in E major, with a brief move towards (what we call a tonicization of) B.

(Note: B major fits just as well as E major, and we could call the A a chromatic pitch in B major. My decision of E major comes from years of experience with tonal music, where it's a very common move for music to begin in the original key and move to the fifth scale degree. This is also more common than the ♭7 that would be the A at the beginning of a B-major excerpt.)

As for transposition, the quickest method (again, just my opinion) is to think in terms of scale degrees. In E major, the pitches you wrote are 1 7 1 7 1 1 7 1 2 3 2 3 4 5 ♯4 5 ♯4 5. Now we can just think in D major—D E F♯ G A B C♯ D—and write out those scale degrees in D to transpose. This results in D C♯ D ♯ D D C♯ D E F♯ E F♯ G A G♯ A G♯ A.

how do I determine which key they are in?

A) Recognise that those notes are the start of the overture to Mozart's The Marriage Of Figaro, but two semitones higher.

B) Note that that piece is in D major.

C) Infer that the notes in the question are therefore in E major.

:-)

More seriously, I don't think you can say definitively just from the notes.  They don't all fall in the same major or minor (or modal) scale, so either are from some bizarre scale, or include some non-scale notes.  (The A#s, if it's really from Figaro.)  Or it splits into sections, from different scales.  (For example, the first 13 notes are from the E major scale, and the rest from the B major scale, which happens to match the harmonies of Figaro.)

how do I determine which key they are in?

1. Learn Major, minor scales and circle of fifth.


2. Then apply the notes into each Key in the circle of fifth.


3. Most notes will fit into one key


4. Most piece will end in key note

In this case all notes(expcet A) fit into Key B Major and the piece ends in B, so this could be most probably in key of B major.

how do I transpose them to a key of D?

1. Find the distance from original key to the new Key in number of semitones.


2. Increment each note by the distance

In this case the difference from B to D is 3 semitones.
And the transposed piece would have the following notes.

G F G F# G G F# G A B A B C D C# D C# D

The piece appears to be in the key of Emaj(1), with a modulation to Bmaj at the end. I speculate this because on the surface it appears that the last 5 notes appear to be a repeat the the first 5, but in the dominant key (i.e. Bmaj(5)). Of course it's impossible to tell without more context.

Taking the Emaj/Bmaj above to be correct for the sake of argument, the transposition to D would be D as the tonic and A as the dominant, hence:

D C# D C# D D C# D E F# E F# G A G# A G# A

Kudos to Richard. Scale degrees is an excellent metaphor for direct transposition.