So to be explicit: I'm assuming that you are in your message app, you click the button to write a new message, type the name in in the field that opens. If you are doing something different (or missing one of those steps) please tell us exactly what. Also what message app (I assume the built-in one, but as you've not said what phone you have that doesn't answer the question - most manufacturers include their own message app).
And when you click on the name and it takes you to your "overall messages", does it take you to a conversation with that person (or at least a list where that person's conversation is visible), or just to the conversation list? The former would make sense if it was treating it as a search for that person (i.e. it didn't recognise that you wanted to send a text, just as if you wanted to find your messages to that person). Of course if you click on that conversation you should be able to send a message directly from there.
Sorry it's still all questions, but there's enough unsaid in your message that I can't be certain of the sequence.
If the problem is the message app misbehaving, have you tried a different message app?