If by correct you mean grammatical, then both sentences are correct. The second utterance (When I met him yesterday, he told me: "I will give you your money tomorrow") reports your friend's exact words - indicated here by the speech marks (i.e. direct speech). The use of direct speech in the spoken reporting of what someone told you, however, is very unusual and needs to have the appropriate prosody to be easily interpreted.
The first utterance (When I met him yesterday, he told me, he will give me my money today) uses indirect / reported speech, which is far more common in spoken language than direct speech. When reporting what was said to them, speakers often backshift the tense of the verb, so that the words "I will give you the money" become: He said he would give me the money.
However, backshift is not mandatory. Very often speakers retain the tense of the verb if they believe that what was said to them still holds true. So, for example, if Friend A told me yesterday "I will give you the money tomorrow" and I see Friend B early in the morning of today (firmly believing that A will keep his promise), then I can say:
When I met him yesterday, he told me he will give me my money today.
Of course, if it is the end of the day, and the probability of getting your money back is now low, then you would use would:
When I met him yesterday, he told me he would give me my money today.
Note: if you are learning this grammar for an exam, you are probably better off sticking with the backshifted version, which some consider to be the default usage in such contexts.
We can put adverbs and adverb phrases at the front, in the middle or at the end of a clause. [Resource from here]
The front position of the clause is the first item in the clause:
Suddenly I felt afraid.
Yesterday detectives arrested a man and a woman in connection with the
murder.
The end position of the clause is the last item in the clause:
Why do you always have to eat so fast?
The mid position is between the subject and the main verb:
Apples always taste best when you pick them straight off the tree.
Where there is more than one verb, mid position means after the first auxiliary verb or after a modal verb:
The government has occasionally been forced to change its mind. (after
the first auxiliary verb)
You can definitely never predict what will happen. (after a modal
verb)
We mightn’t ever have met. (after the modal verb and before the
auxiliary verb)
In questions, the mid position is between the subject and the main verb:
Do you ever think about living there?
Adverbs usually come after the main verb be, except in emphatic clauses:
She’s always late for everything.
When be is emphasized, the adverb comes before the verb:
Why should I have gone to see Madonna? I never was a fan of hers.
(emphatic)
Best Answer
Both are correct. One is more clear.
The concept of "when to talk" and "what to talk about" are both adverbs modifying the verb "talk". If you put the "when" after the "Travel Plans" you introduce some uncertainty, because the travel plans could be made more specific by including a time that the plan is for. Maybe you're modifying the travel plans with "when".
You could eliminate that uncertainty by putting the "when" after "Talk" rather than "Travel plans".
So "I want to talk to you later today about my travel plan." is less ambiguous.