Tempo is traditionally indicated in Italian, and many classical pieces are named simply by their tempo. Common indicators include "allegro" (fast), "allegretto" (slightly slower than "allegro"), "presto" (very fast), "prestissimo" (extremely fast), "vivace" (lively and brisk), and "vivacissimo" (very lively and brisk). You may also encounter modifiers to these terms, such as "allegro na non troppo" (fast, but not so much), or "presto con fuoco" (very fast, with fire).
The Harvard Dictionary of Music takes note of "the standard formal cycle of the early symphony, three movements in a fast-slow-fast pattern" and the subsequent "expansion of the symphonic cycle from three movements to four." Virtually any late 18th or mid-19th century symphony, sonata or concerto will begin and end with a movement in fast tempo, although symphonies in particular may first contain a brief, slow introduction. The symphonies of Mozart and Haydn best exemplify this convention.
You can also expect any baroque or pre-classical work bearing certain titles to have a fast, lively tempo. A "toccata" is a work for keyboard instrument designed to feature rapid, virtuosic playing. Likewise, the "gigue," "bourree," and "rigaudon" are all lively French dances favored by Baroque composers such as Bach and Handel.