The metre carries the tempo
For the two centuries before the metronome, the time signature itself was half the tempo instruction. A player read the metre, the smallest note values, and the word together, and arrived at a speed. The habit still works, and it still matters — because most of the music this course is heading toward was written that way.
The metre is part of the speed
Cut time (₵) against common time (C). Same notes on the page, but ₵ is counted in two half-note beats, not four quarters — so the bar goes past about twice as fast.
The compound bar has a longer stride
An Allegro in 6/8 is quicker than an Allegro in 3/4. The 6/8 bar is felt in two big beats, each carrying three eighths — so "one bar" covers more ground, and the piece pushes forward. The finale of the Mozart concerto you're aiming at is a 6/8-feel Rondeau: Allegro, and it bowls along for exactly this reason.
Small notes, slower tempo
A piece packed with sixteenth notes is played more slowly than one built from quarters — even under the same word. The busy values fill the time; pushing the tempo would turn them to mush. This implied "proper speed" has an old name: tempo giusto, the tempo the notation itself asks for. Before you set a number, look at the fastest notes on the page.
Try it
Try
Pick two pieces you know. For each, note the time signature and the smallest note value on the page, then guess the tempo before playing. A 6/8 with long dotted values wants a flowing two-beat pulse; a 4/4 full of sixteenths wants patience. Check your guess against a recording.
The usual mistake
Reading only the word and ignoring the metre — so a cut-time march gets played as slowly as a common-time one, or a 6/8 Allegro gets raced because "Allegro means fast." And the across-the-board one: a Classical Allegro is contained — brisk, articulate, unhurried — not the headlong tempo a modern ear expects. When the metre and the small notes say "steady," believe them over the word.