2 to publish abroad, declare
Translated preach-1, signify-1, declare-1

diagellomai

1. to think different things with one's self, mingle thought with thought
   1. to ponder, revolve in mind
2. to converse, discourse with one, argue, discuss
   Translated dispute 6, reason with 2, reason 2, preach unto 1, preach 1, speak 1

euaggelizo 

1. to bring good news, to announce glad tidings
2. used in the OT of any kind of good news of the joyful tidings of God's kindness, in particular, of the Messianic blessings
3. in the NT used especially of the glad tidings of the coming kingdom of God, and of the salvation to be obtained in it through Christ, and of what relates to this salvation
4. glad tidings are brought to one, one has glad tidings proclaimed to him
5. to proclaim glad tidings
6. instruct (men) concerning the things that pertain to Christian salvation
   Translated preach 23, preach the Gospel 22, bring good tidings 2, show glad tidings 2, bring glad tidings 1, declare 1, declare glad tidings 1, miscellaneous 3

katagello

1. to announce, declare, promulgate, make known
2. to proclaim publicly, publish
3. to denounce, report, betray
   Translated preach 10, show 3, declare 2, teach 1, speak 1
   

kerusso

1. to be a herald, to officiate as a herald
2. to proclaim after the manner of a herald
3. always with the suggestion of formality, gravity and an authority which must be listened to and obeyed
4. to publish, proclaim openly: something which has been done
5. used of the public proclamation of the gospel and matters pertaining to it, made by John the Baptist, by Jesus, by the apostles and other Christian teachers
   Translated preach 51, publish 5, proclaim 2, preached + 2, preacher 1

laleo 

1. to utter a voice or emit a sound
2. to speak
   1. to use the tongue or the faculty of speech
   2. to utter articulate sounds
3. to talk
4. to utter, tell
5. to use words in order to declare one's mind and disclose one's thoughts
   1. to speak
      Translated speak 244, say 15, tell 12, talk 11, preach 6, utter 4, miscellaneous 3,  speak 1

parrhesiazomai

1. to use freedom in speaking, be free spoken
   1. to speak freely
2. to grow confident, have boldness, show assurance, assume a bold bearing
   Translated speak boldly 4, preach boldly 1, be bold 1, wax bold 1, boldly 1, freely 1

proeuaggelizomai 

1. to announce or promise glad tidings beforehand
   Translated preach before the gospel 1

prokerusso

1. to announce or proclaim be herald beforehand
2. to announce beforehand (of the herald himself)
   Translated preach before 1, preach first 1

We are all preachers, not necessarily from a pulpit, but preachers of the Word of God and teachers help us to learn the word of God to share with a lost world.

I think the problem is we are trying to separate two things which aren't completely separate. it is here in this passage that we find both preaching and teaching. I believe the going and making disciples is indeed the aspect related to preaching, but isn't it also related to teaching. Even today Christians don't understand just how lost we were or how heinous sin is to God. Nevertheless Jesus connects it to teaching. So aren't we forced to acknowledge at least, that there is a point where the two cross over into each other. 

PREACH THE GOSPEL & TEACH THE SAINTS

Thank you to the person who outlaid the Greek on this. That is correct. Biblically, preaching is a public proclamation and teaching is for the church. However, often in common use today the two are confused. Some think that a sermon is for preaching and a Bible study is for teaching, but that was not the original meaning. That is OUR modern practice, but not what the Bible means by preach and teach.

Some people think that exciting and dramatic teaching which is just for the church is preaching, and that dull and lifeless spreading of the gospel is teaching. None of the message of God ought to be dull and lifeless, but on the other hand, sometimes excitement is a cover up for a lack of depth and content.

Originally, it was simply preach the gospel (in public, outside of the church) and teach the saints (after baptism, inside of the church). It originally had absolutely nothing to do with the modern practice of "preaching" a sermon.