Roots of the difference between 谈,论,说,讲,议, 叙,言,报,告,述
言 (yan2) is the root of all words meaning talk. It says so in the 說文解字 (a dictionary from the Han dynasty):


  凡言之屬皆从言


The origin of the character 言 is a picture of a person with a big mouth. In ancient Chinese, it’s the general word for any form of speech or talking. In modern Chinese, it has become literary and is normally only used in compound words like 語言 (yu2yan2) "language".

Most other characters you mention are just 言 with a second component to indicate the sound of the word (this type of character is called 形聲字 (xing2sheng1zi4):

談

(tan2)’s phonetic part is 炎 (yan2), which means hot or inflamed. In ancient Chinese, this word has connotations of talking happily about something. It’s often used in the combination 談笑 (tan2xiao4), meaning "to talk and laugh". In modern Chinese, this is a general word for light conversation, similar to English "chat" or Russian "razgovor".

論

(lun4)’s phonetic part is 侖 (lun2), meaning logical order. In this case, this part also points to the meaning: Both in ancient and modern Chinese, 論 means to explain or to discuss. It can also mean to ponder something without actually talking out loud. Example from the 呂氏春秋:

闻言必熟论 the words heard must be carefully examined


說

(shuo1)’s phonetic part is 兌 (dui4), which can mean a number of things such as "exchange" or "melt". In ancient Chinese, 說 has two meanings. Pronounced "shui4", it means "to persuade". Pronounced "shuo1", it means "to explain", or, as a noun, "teachings". For example, 老子說 means "the teachings of Laozi". The meaning "explain" still exists in modern Chinese, but 說 has also become a general word for "speak", like English "say" or Russian "skazat". 

講

(jiang3)'s traditional character had the phonetic 冓 (gou1), which can mean a number of things, including "to meet" and "to combine". Accordingly, 講 in the 說文解字 means "to reconcile". The simplified character uses 井 (jing3), meaning water well (Russian "kolodets"). In modern Chinese, 講 has a number of meanings, but the most common ones are:

講故事 tell a story
講價 negotiate the price


In some dialects, 講 replaces 說 as the main verb for "speak".

議

(yi4) has 義 (yi4), which means "meaning" or "righteousness". Both words reflect the meaning of 議, which means "opinion" or "comment" and is normally used in compound words.

告 (edited)

There are different explanations of this character. I originally followed the "cow that makes a noise" explanation found here, but that seems to be wrong.

告 (gao4) consists of the parts 牛 (niu2) "cow" and 口 (kou3) "mouth". 
Various explanations for this character exist. The 說文解字 explanation does not make sense to me. This article (PDF in Chinese) claims that the character originally described a method of catching or keeping cows. Later it acquired the sense of "to sacrifice" (also written as 祰). From there, its meaning as "report" developed. In ancient Chinese, 告 meant "to report something to a superior". Nowadays it means "to tell somebody about something".

報

(bao4) is interesting. The left side is a criminal (辠 zui4). The right part is a hand and a kneeling person, a shorter version of the character 服 (fu4) "punish". 報 originally meant "court documents", but the meaning has expanded to cover the English words "report", "answer", "repay" and, from those meanings, "newspaper".