First, there is a distinction between monastic or regular clergy, who lived according to a communal rule (regula), and secular clergy, who formed the majority of the clergy who ministered in parishes. Regular clergy had distinctive dress (formalized over time) so that, for instance, Franciscans friars wore brown or gray habits (and eventually were nicknamed “Gray Friars”) and Dominican friars wore a black over-cloak, hence “Black Friars.” Pope Innocent III’s great Fourth Lateran Council of 1215 enacted a canon (number 16) regulating the clothing of the secular clergy (and there were earlier version of this in various regional councils). It didn’t prescribe a standard clerical uniform as much at it outlawed clothing that was deemed unsuitable for the clergy. It said:

Their [the clergy] outer garments should be closed and neither too short nor too long. They should not wear red or green fabric, long sleeves or shoes with embroidery or pointed toes, or use bridles, saddles, breast-plates and spurs that are gilded or have other superfluous ornamentation. . . They are not to wear buckles or belts ornamented with gold or silver, or even rings except for those whose dignity it befits to have them.

With local variations, this canon was regularly repeated in provincial church councils and synods. Thus, in his provincial constitutions issued in 1342, the Archbishop of Canterbury, John Stratford, embellished Innocent’s canon, noting that:

Exterior dress often displays the inner character and condition of people; and though the dress of the clergy should be an example and instruction for lay people, the abuses of clergymen in their tonsures, clothing, the trappings of their horses, and other things, which these days are on the increase more than usual, has created an abominable scandal among the people. . . .[Men in holy orders] walk about in military rather than clerical attire, wearing a short but excessively full surcoat with long sleeves not reaching to the forearms, but hanging with tails, lined and wrapped with fur or horsecloth . . . and capes with marvelously long tippets (hoods), and with long beards, publicly wearing rings on their fingers without distinction, and girded with expensive studded belts of amazingly large size, and purses enameled with pictures and gilded with various embossings, from which hang small knives like swords, also with red or green checkered shoes or shoes pointed and incised in various ways. [My translation]

In short, priests were expected to wear clothing in subdued colors (not necessarily black) that was ankle length. The white clerical collar that we know is really a 19th-century invention though there are versions of it, i.e., various kinds of white cravats, earlier on.

Two further points: All men in clerical orders—regular and secular—were expected to wear a tonsure, i.e., a shaved portion of their scalp (Lat. “tondere,” to shave or shear). What part of the scalp was shaved varied over time and place but generally it was the crown of the head. This was the distinguishing mark of clerical status.

All priests wore special vestments when celebrating mass. Wikipedia is the handiest place for a description of the various articles of clothing that formed a suit of vestments.

Finally, ordinances meant to regulate the clothing of the secular clergy were frequently repeated in the Middle Ages, suggesting that not every parish priest paid strict attention to them.

TL;DR: There was no standard outfit for ordinary medieval priests. You were supposed to be able to spot a member of the clergy by their subdued clothing and their tonsure.

For more detail see Sarah Hamilton, Church and People in the Medieval West, 900-1200 (2013), p. 70 ff. Google books snippet here

This is a book response that I think can help shed light on your question, as well as elaborate on the "official" response of /u/Whoosier .

Maureen Miller’s book, Clothing the Clergy is a chronological study between the years 800-1200. Miller makes connections back to Roman traditions. At the basis of her study, she wanted to understand how and why so much importance was put on clerical clothing, and what the effects of this was economically, ideologically, and emotionally. Her study comprised the regions of Italy, England, and the lands conquered by the Carolingians.

To start, she details major changes in the 11th and 12th centuries which focused on legislation and practices of clerical garments inside and outside of the sanctified space. The introduction of penalties as a result of these changes highlighted the newly prescribed importance in clerical culture for these distinctions. As an addition, an increased desire to differentiate clerics from laymen grew. The types of liturgical garb at the time differentiated because of the need for different types of garb inside and outside of ceremony. Miller traces these differences back to Roman Tunics, and the advent of toleration under Constantine in 313. This toleration led to a period of experimentation by Roman religious groups and led to differentiation in garb and colors to identify religiosity. As a result of these differentiations and changes, there was a division of views on garments exemplified by passages from Celestius. The want of priests to identify themselves as educated and separate led to growth in philosopher priests who used special garments left over and bestowed upon them as an illustration of their education and virtue. Miller uses other examples of clerical culture responding to changes and criticisms of Western society as catalysts of evolution and change of clerical garb. These ranged from

the introduction of the Tonsura and Corona, the bestowal of the pallium, and the permission by popes to wear special roman garments as a way to create closer ties in ecclesiastical circles.

As a result of these increasingly accepted and revolutionizing changes, texts and practices such as the Pontifical combined older Roman rites to newer Frankish usages. The dissemination of these texts, in addition to new Papal councils in the 12th century, allowed for a greater unity of ideology and usage of vestments. Where in 1089 at the Council of Melfi and the Fourth Lateran Council, we have a concrete look at official doctrine On the Dress of Clerics. Outlining punishments for not wearing proper attire, and having certain pieces classified as excessive or lay garments. This led the clergy to create new garments and modify older ones. In an attempt to educate the clergy, Pope Gregory spread Pastoral Care which detailed sacred vestments bestowed by God on to Aaron, and attributed allegorical meanings to garb and to use symbolism as a tool of reform.

The allegorization of vestments was seen as a new weapon by the clergy. Since clergy were meant to be men of peace, they used these self-proclaimed and spread powers as a way to separate themselves from secular power. The involvement of Bishops with Kings led to an increased display of ornamental attire, where they believed “Earthly rulership is a joint exercise between kings and Bishops”. The increasing display of ornamental garb led to a pushback by lordly rulers and issuing of delimitations. These delimitations only furthered the clergy’s usage of their spiritual virtues under their allegorical garb, and led to further threatening of secular rule with excommunication. Eventually leading to patrimonial stewardship and the molding of secular rulers to ecclesiastical ends.

I'll format and edit this a little better in a bit