A Beginner's Guide

Eighteenth-century students learned partimenti at the keyboard, and today that is still the best method. You do not need to be an accomplished keyboard player. It is enough to be able to play, slowly and at your own tempo, a one-note bass in the left hand and one, two, or three notes in the right hand. You will be training your ears more than your fingers.
Start by learning the "Rule of the Octave," or as the Italians called it, the regola dell'ottava. By that they meant the most common chords to play over each note of the the ascending and descending scales. An illustrated guide to the "topography" of the Rule of the Octave can be found by clicking here. You can find descriptions of how to perform the Rule of the Octave under the headings Hand Positions in the rule books or regole of Fedele Fenaroli and Giovanni Furno. Learn these patterns in all the major and minor keys, at least those having up to three or four sharps or flats.
The Rule of the Octave, as depicted in the rule books, demonstrates sonorities, not the movements of four or even five independent voices. That is, do not attempt to apply the rules of part writing to the inner parts of the Rule of the Octave. Think of it more like playing a reduction of an orchestral score, where a number of parts may be doubled in other octaves. As you progress to real partimenti, thin the texture to two or three parts (including the bass) and begin to think of them as real voices.
Work your way through the rule books, playing each musical example in several keys, both major and minor. Furno's and Durante's regole contain small partimenti suitable for beginners. Your central task will be to study each partimento to see where you might employ three types of patterns: (1) segments of the Rule of the Octave, (2) cadences, as specified in the regole, and (3) special moves or movimenti either required or optional for various sequences. If you find yourself thoroughly perplexed by, for instance, one of Furno's partimenti, you may wish to consult the editor's sample realizations found in the Appendix to that treatise. This should be a last resort.
Even simple partimenti change key frequently. Often you may be able to change keys through one of the three types of patterns mentioned above. Try to know the scale degrees of the three or four upcoming tones in the bass. When there is a conflict between the old key and the new key, favor the new key.
As you become more comfortable with simple partimenti, begin to learn the typical right-hand patterns of decoration laid out in Franceso Durante's Embellished Basses, or Partimenti diminuiti. In imagining an ideal texture, think of Domenical Scarlatti's harpsichord sonatas, not J. S. Bach's chorales.
A variety of small partimenti, many with chromatic passages, can be found in Mattei's Piccolo Basso ("Little Bass"). Each poses an interesting musical problem for the performer. Their brevity, however, helps to limit their difficulty. They also provide practice in reading more complex figures.
As you experiment with various partimenti on this website, you may begin to notice passages where the bass performs a motive and then pauses on a long tone. This is often a cue to repeat that motive in the melody. The higher levels of partimenti presume a polyphonic give and take between the bass and melody.
Among the most advanced partimenti are the partimento fugues. Though we have relatively little historical information about how these works were performed by students, the best advice may be to keep the texture thin. Do not attempt to perform a four-voice fugue with all four voices sounding all the time. This is bad practice even in written-out fugues, and almost surely fatal for improvised partimento fugues.