In modern English, express is a verb meaning – broadly – to convey information, generally of a more personal nature, as well as an adjective meaning very fast. From the verb, we have formulas such as ‘expression of regret’, ‘self-expression’, ‘expressing milk’, whilst we get ‘express train’ and ‘express delivery’ from the adjective.
The word originally stems from the Latin exprimere, meaning ‘to push or press out’.
Italian coffee machines force the coffee out under high pressure, hence espresso (past participle of the verb esprimere): nothing to do with speed, except they produce coffee faster.
In French, a train which doesn’t stop at all the stations is an exprès, meaning ‘deliberate’, as in the English ‘my express intention’: nothing to do with speed, really, except it gets you there faster.
English has taken both espresso and exprès and re-interpreted them to mean ‘fast’.
Expressing milk retains the etymological sense of pushing out (in fact, breast feeding is more of a question of pulling, since it works by creating a lower pressure outside the nipple than inside the breast).
Considered somewhat obliquely, ‘to express’ in the sense of ‘to share information’ is also a question of ‘pushing something out’, from our minds, from our mouths. Many languages have a calque of this word, such as the German ausdrucken, a combination of aus (out of) and druck(en) (pressure, press), or the Russian выражение (vyrazhenie) from вы = out of and ражение = striking.
It will come as no surprise that ‘impress’ and ‘impression’ have similar backgrounds, but that’s another story.