We know quite a lot about Mozart's playing, much more than, say, J. S. Bach, because he was extremely famous in the courts of Europe, and those people left plenty of accounts behind.

In January of 1781, the Emperor of Austria, Josef II, hosted a competition between Mozart and Muzio Clementi. From Harold Schonberg's "The Great Pianists":

Clementi led off with an improvised prelude and his Sonata in B flat (Op. 47, No. 2) [...] Then Clementi followed with one of his specialties, a toccata featuring thirds and other double notes. Now it was Mozart's turn. He too improvised a prelude, and followed with a set of variations. The Grand Duchess produced some sonatas of Paisiello [...] and both pianists read them off at sight, Mozart playing the allegros, Clementi the adagios and rondos. Both were asked to select a theme from one of these sonatas, developing it on two pianos. Presumably Mozart would have taken a theme and played it, Clementi noting the harmonies. Then Clementi would have accompanied Mozart on the second piano while Mozart developed his material. And vice versa. It probably ended with a grand two-piano splash, in which all the melodic fragments were woven together.

Victory was left undecided [...] A few years later, the eminent composer Karl Ditters von Dittersdorf happened to be talking to Joseph II, and reported the following conversation in his autobiography:

EMPEROR: Have you heard Mozart play?

I: Three times already.

EMPEROR: How do you like him?

I: As any connoisseur must like him.

EMPEROR: Have you heard Clementi?

I: Yes, I have.

EMPEROR: Some people prefer him to Mozart, among them Greybig ["Greybig was Franz Kreibach, violinist and conductor in the Emperor's orchestra] What is your opinion? Be honest.

I: Clementi's way of playing is art alone. Mozart's is art and taste.

EMPEROR: That's just what I said...

In the 18th century, all keyboard composers were accomplished players, but they were evaluated differently than modern performers. More than mechanical virtuosity, they were noted for their ability to improvise and for the originality of their own style and compositions.

Ralph Kirkpatrick in "Domenico Scarlatti" writes:

[F]ew keyboard players possessed a universal technique that permitted them to negotiate immediately any music set before them. Even the greatests players were accustomed to perform only their own music or the music of their own country or school. [...] Only when keyboard playing became a profession in itself as apart from improvisation and composititon did a genuinely universal technique such as that exemplified in the miano methods of Hummel and Czerny become standard equipment even for players of only average ability.

You might enjoy reading Mozart's letters, which sometimes included his thoughts on music. He was a consummate sight reader, the key to playing any instrument well, and he conducted his orchestra while playing violin or piano when playing his concertos. He made a remark in his letters about the importance of playing legato properly, that it should "flow like oil." (He also referred to Clementi rather amusingly as a "mechanicus.") Playing classical piano or other instrument is all about sight reading and has nothing to do with any style other than the composer's, by the way. I suggest taking in some of Mozart's later piano concertos, perhaps the 19th, 29th, 21st, or 22nd. Remember as you listen that very shortly before he came of age, harmony was not played. His compositions absolutely mastered it.