(PW) make a bad start

Having a fight with a co-worker on my first day of work was not a good idea.  I got off on the wrong foot.

(free)

on the right foot

In an auspicious manner: The project started off on the right foot but soon ran into difficulties.

on the wrong foot

In an inauspicious manner: The project started off on the wrong foot.

(GoEnglish)

Get Off On The Wrong Foot (have a bad start; start badly; the two did not like each other when they met ... )

(Phrases.org) Make a bad start to a project or relationship.

This has the sound of an old expression - from Shakespeare, the Bible or similar. Shakespeare did use the notion of a 'better' foot (which implies a wrong foot) in King John, 1595:

KING JOHN:
Nay, but make haste; the better foot before.
O, let me have no subject enemies,
When adverse foreigners affright my towns
With dreadful pomp of stout invasion!
Be Mercury, set feathers to thy heels,
And fly like thought from them to me again.

Richard Harvey, in Plaine Perceuall the peace-maker of England, 1590, is the first to record the wrong foot in print:

"Thou putst the wrong foote before."

Despite the implication otherwise in the phrase put your best foot forward we only have two choices, so if there's a wrong foot there has to be a right one too and get off on the right foot is also in common use.

How did these phrases originate? Well, we don't know. It may be that it comes from the long-standing preference people have for the right. Most people in all cultures are right-handed and in English at least the bias is part of the language. We have right and left and right and wrong, tends to associate left and wrong. That association is built into the language in the way that we have taken the Latin for left - sinister, to mean dark and suspicious. There are various disparaging terms for use of the left that demonstrate this bias - cack-handed, goofy-footed etc.

There is a suggestion that it in ancient Greece it was considered unlucky to put the left foot on to the floor, or into one's shoe, first. Brewer records this in his Dictionary of Phrase and Fable, 1898. I can't find supporting evidence for that view, so I'll just repeat what Brewer had to say here:

It was thought unlucky to enter a house or to leave one’s chamber left foot foremost. Augustus was very superstitious on this point. Pythagoras taught that it is necessary to put the shoe on the right foot first. "When stretching forth your feet to have your sandals put on, first extend your right foot" (Protreptics of Iamblichus, symbol xii.). Iamblichus tells us this symbolised that man’s first duty is reverence to the gods.

Another suggestion is that the concept of a right foot and a wrong foot comes from the military, where in order to march in step soldiers all have to start with the same foot.