Upon looking for a church, I came across one that does a "three-fold communion". They claimed Jesus commanded everyone needs to wash their feet before eating the bread and drinking the wine, citing John 13:14-15 as proof. Is that a correct interpretation of the passage? If not, should I avoid that church?
The only aspects of the communion observance which are mandated are those recorded in 1 Corinthians 11:23-29:
Apart from the requirements of that passage, the church has liberty to alter the celebration in a variety of ways, including adding the washing of feet, if desired.
On the other hand, these additional observations are not mandated by Scripture, and therefore we ought not teach them as such. In the case of washing feet prior to taking communion, this practice is not prescribed for the Church in Scripture, and John 13:14-15 is not an appropriate proof text for defending such a mandate.
First, Jesus used the moment of washing feet in John 13 to illustrate a larger point that His disciples must maintain a servant attitude of humility. Washing feet was a practice of that day related to traveling by foot on dirt roads. Prior to every meal, it was customary for the host of the home to have the feet of guests washed by a servant, but Jesus surprises His disciples by assuming the posture of a servant to them. He does this to make a point.
Notice in v.15 Jesus called that moment "an example," and examples are given to illustrate larger principles. Jesus calls us to follow the principle behind the example, not merely the specific illustration itself. Therefore, Jesus wasn't commanding us to repeat the ritual of feet washing; He commanded us to maintain an attitude of humility with one another which was exemplified by the ritual.
So today if a church practices feet washing as part of its communion celebration, this is not wrong, however if they say such a practice is required by Scripture, they have misinterpreted the Scripture and crossed over into legalism. Since every church has its flaws, only you can decide if this legalistic practice is enough to warrant fellowshipping elsewhere.