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I have been told when composing the first thing is to establish the key. Beethoven's fifth symphony in c minor starts with major third interval. "...-". I would expect a major key. What am I missing?

Music is an art, and although every art has its own logic, the artfulness comes to bear when a creator marshals that logic toward some more emotional end. You correctly identify the first melodic interval that sounds in this symphony as a major third, and it’s true that absent any other context that interval would suggest that the key is Eb major. But! Further developments in the musical texture soon clarify the fact that the governing key is, in fact, C minor, and it’s precisely that initial ambiguity that marks Beethoven’s artful use of the means of tonal music. In fact, it’s a pretty straightforward piece in that regard: it’s only a few measures before it becomes completely clear that you’ve heard the third and fifth of a minor scale rather than the first and third of a major scale, and so in that sense Beethoven is not diverging too far from the guideline that you’ve cited. The takeaway, I guess, is that music is the art of managing expectations. You can use those expectations that we all kind of share to create a particular implication of what is to follow, and then create surprise by recontextualizing those notes in a different tonal environment. I’m so used to that piece that I can’t help but hear the initial tones in terms of the eventual minor key, but it must be a beautiful thing to hear it for the first time and reckon with the surprise of what follows.

Well, you've been told wrong, haven't you! The first thing Beethoven does is establish one of the themes that he's going to develop in the piece. Development of what may have seemed a simple musical fragment is the basis of the 'classical' style that Beethoven epitomises.

In this case, Beethoven is delighting in being ambiguous. The first four notes of the 5th could be in Eb major. The next four notes do nothing to contradict this. But the next six firmly change the context to C minor. A bit of 'development' already! Clever, isn't it!

Maybe there's a more basic misapprehension in the original question though. It opens with a major third rooted on Eb, not a 'C major interval'. Not every interval in a minor key or scale is minor. A minor triad, for instance, is a major third on top of a minor third.

It's true that the beginning is (probably deliberately) ambiguous, however I would argue that even the first four bars do point to c-minor rather than E♭ major.

If it were E♭, then the very first note would be a standalone third. Now that's perhaps not something you can't do, however the third of the tonic doesn't ever really have any strong dramatic character – it's the harmonious addition note which shapes the character once you have a clear tonic that's already singled out by something else. But here, the very first note is anything but a shy accompaniment voice – it's a strong, determined assertion, so we're in either tonic-fundamental or dominant territory. In fact the first three notes might most reasonably be heard as the tonic, pointing us to either G-major or g-minor.

X:1
L:1/8
M:C
K:G
%%score T1
V:T1 clef=treble
% 1
[V:T1] (3GGG "^?" [GBdg][Dcdf] "^?" [G,Bdg]2

But then there's the E♭. If we were in G-major, that would be a chromatic mediant, which would be a bit far-fetched at this point. g-minor then, perhaps? Then the E♭ would be either a predominant Ⅵ root or ⅳ third, or the ♭9 of a diminished-seventh chord that would lead us straight back to g. In either case, we would in such a dramatic classical setting expect to find a clear leading tone of the dominant soon, i.e. an F♯.

X:1
L:1/8
M:3/4
K:Gm
%%score T1
V:T1 clef=treble
% 1
[V:T1] (3GGG E2 "^?" ([D^F][CF]) "^?" | ([B,G]G,)

Well, the next note is F though. Natural. Ok, that does occur in the g natural-minor scale, but it doesn't in any way fit in the dramatic mood if it's just a Ⅶ degree of g-minor. Confusion at this point.

Then there's the D. We would have expected this as the dominant to g, but then why the hesitant F-natural? This wouldn't make sense. And it didn't sound hesitant, no, this was a strong note like the others. If we have an assertive G in the room but now also an F, then this actually seems to be a seventh chord that's implied – a dominant-seventh G chord. And this leads to either C-major or c-minor. We've already seen an E♭, so probably C-minor.

X:1
L:1/8
M:3/4
K:Cm
%%score T1
V:T1 clef=treble
% 1
[V:T1] (3GGG HE4 | (3FFF "!" H[G,DF=B]4 | [C,CEc]6

That's just strange rhythm-wise, why would he make the dominant clear only on the 2 beat like that? No it's not a Sarabande, it just turns out the rhythm is different from what it sounds like and the triplets aren't actually triplets.

X:1
L:1/8
M:2/4
K:Cm
%%score T1
V:T1 clef=treble
% 1
[V:T1] zGGG | HE4 | zFFF | HD4

Both major and minor triads contain a major third and a minor third stacked in some order. Going up from the root, a major triad has a major third and then minor, while a minor triad has a minor third and then major.

Beethoven's Fifth Symphony leads off with the two notes of the major third that forms the top of a C minor triad (G and Eb). Those two pitches could be the bottom two notes of an Eb major triad (Eb-G-B), or the top two notes of a C minor triad (C-Eb-G). To establish whether a triad is major or minor, one either needs to have three notes, or have enough context to establish a key center. The very start of Beethoven's Fifth provides neither.