I'm wondering if there are (or were) any Christian communities or traditions that pay (or paid) more attention to the teaching of Jesus than to the teaching of apostles.
The various "New Church" denominations that follow the teachings and Bible interpretations written by Emanuel Swedenborg (1688-1772) give much greater weight to the Gospels (and the book of Revelation) than to the Acts and the Epistles.
As explained in my answer to the question, "What writings are held as 'biblical canon' by Swedenborgians?," in the New Testament, Swedenborg viewed only the four Gospels and the book of Revelation as part of the inspired Word of God. The remaining books in the New Testament he said were "good books of the church," but inspired more in the way that the Holy Spirit inspires a preacher when preaching based on the Word of God.
Because of this belief, Swedenborg based his Christian theology primarily on the Gospels and the book of Revelation (in the New Testament), and only secondarily on the Acts and the Epistles. Swedenborgians have generally followed suit.
Having said that, both Swedenborg himself and various Swedenborgian theologians do also quote from the Acts and the Epistles in supporting and explaining their theology. Swedenborg stated that these books, when rightly understood, also support his theology.
From the Catechism of the Catholic Church, there's a pretty clear statement that the Gospels hold the highest place.
CCC 125 The Gospels are the heart of all the Scriptures "because they are our principal source for the life and teaching of the Incarnate Word, our Saviour".
CCC 127 The fourfold Gospel holds a unique place in the Church, as is evident both in the veneration which the liturgy accords it and in the surpassing attraction it has exercised on the saints at all times:
- There is no doctrine which could be better, more precious and more splendid than the text of the Gospel. Behold and retain what our Lord and Master, Christ, has taught by his words and accomplished by his deeds. (St. Caesaria the Younger)
- But above all it's the gospels that occupy my mind when I'm at prayer; my poor soul has so many needs, and yet this is the one thing needful. I'm always finding fresh lights there; hidden meanings which had meant nothing to me hitherto. (St. Therese of Lisieux)
Based on what I have observed with current practice, the Roman Catholic Church puts the most weight on the Gospels, in terms of the books in the New Testament. This can be seen during the celebration of the Mass during the Liturgy of the Word.
You can see it's order here (pages 46 and 47) in the the 3d Edition Roman Misal.
Before Mass begins, during the procession, the book that holds the Gospels is carried in by a Deacon or Priest, held high above his head for all to see.
During the Liturgy of the Word, there are first readings by lectors (usually lay persons) from elsewhere in the bible:
Old testament, a responsorial Psalm, an Epistle.
There is then a profound pause.
A Hallelujah is sung and the people stand.
A Priest or Deacon proclaims the Gospel. ("A reading from the Gospel according to Luke," for example).
The people cross their heads, lips, and hearts and say
"Glory to you, O' Lord."
The Gospel is read.
The Priest or Deacon then concludes with
"The Gospel of the Lord"
The people respond
"Praise to you Lord, Jesus Christ."
The people then sit and listen to the homily.
The homily that follows is most often focused on the teachings of Christ, with supporting points woven in from the other readings.
Conclusion: It is fair to say that, in theory and in practice, the Roman Catholic Church presents the Gospel as the most important part of the New Testament.