Let me try:
言う： The only thing on your list that I'd actually translate as "to say" or "to tell". This is sort of "neutral" in that it really just means to say. For example:
「彼女がいない」と言ったでしょう？ = He doesn't have a girlfriend, I told you so already!
Note that as I am not giving any context you could equivalently translate the sentence as: He said he didn't have a girlfriend, or did he not!
述べる： Can mean "to express" or "to state":
英語で自分の考えを述べるのは難しい。= It is difficult to express one's thoughts in English.
(Example sentence taken from jisho.org)
話す："to talk", as you wrote above. For example:
今お母さんと話しています。= I'm talking to mum right now.
おしゃべりな人と話したくない。= I don't want to talk to talkative people.
伝える： "to convey" as in "Say hello to X.":
田中さんにこの知らせを伝えて下さい。= Please pass these new onto Mr. Tanaka.
告げる： I've not heard this one before (which doesn't mean much) but it seems to mean "to tell" or "to convey" again. Examples (again taken from jisho.org):
もし私がそれを知っているなら、私はそれをあなたに告げるだろう。= If I knew it, I would tell you.
教える： Can mean "to teach" or "to tell". For example:
あなたの電話番号を教えて下さい。= Can you give me your phone number?
彼は日本語学校で日本語を教えています。= He's teaching Japanese at a Japanese language school.
This answer is incomplete but I hope this helps.