German cursive means two things: the school scripts taught in Germany since the 1950s, and the older Kurrent and Sütterlin hands in which letters and records were written until the 1940s and which most Germans can no longer read.
The school scripts
West Germany introduced the Lateinische Ausgangsschrift in 1953, a looped, right-slanted Latin cursive; East Germany introduced its own Schulausgangsschrift in 1968; and the Vereinfachte Ausgangsschrift of 1972 simplified the joins so that every lowercase letter ends at the midline. All three are fully joined, with looped ascenders and descenders and entry strokes on the lowercase. Which one a child learns depends on the federal state and the school, and some states now teach a basic unjoined script first. This site renders the model in a German school face.
Kurrent and Sütterlin
Before 1941 German was written in Kurrent, a pointed, angular cursive descended from medieval hands, with letterforms quite unlike the Latin ones: e is a pair of short strokes, s has a long form and a round form, and n and u look alike. Sütterlin, introduced in Prussian schools in 1915, is a simplified upright Kurrent. Both were dropped by decree in 1941 and replaced by Latin script, so letters, diaries and records from before that date are in a hand that later generations were never taught. Reading them is a skill of its own.
Reading old German handwriting
Learn the Kurrent alphabet first, lowercase then capitals, and expect to read by word rather than by letter, since several letters are told apart only by context. Fraktur, the printed blackletter, is a different thing from Kurrent, the handwritten form, and is easier to read. The reading old cursive guide covers the Latin hands; genealogical societies publish Kurrent alphabets for the German ones.
Practising it
For a child in a German school, the school's own Ausgangsschrift and its ruled exercise books. The worksheet generator prints the looped model on standard rules, and the looped cursive page describes the family it belongs to.