I and my thoughts on website www.___.com
Me and my thoughts on website www.___.com
Which one of these phrases is correct?
As sentence fragments, they're both absolutely fine (though, I think you need to put a definite article in front of the word website: on the website www.___.com). That's the short answer.
The long answer is whether they're correct or not depends on where in a sentence (a sentence consists of a subject and a predicate) they're going to be found. If at the beginning, then only I and my thoughts would be correct as I is a subject pronoun while me is an object one, which means that it cannot be placed before the predicate as it is the place, at least in English, where only the subject of a sentence is found. Subject pronouns (I, you, he, she, it, we and they) are used in places where we expect to see the subject of a sentence. Likewise, objects pronoun (me, you, him, her, it, us and them) are used in places where we expect to see objects. An object is a noun or noun phrase governed by an active transitive verb or by a preposition. For example:
I and my wife are going to visit Paris this summer.
We use I here because, as we know, the subject of a sentence only works with subject pronouns.
They gave me and my wife a free ride to the city center.
In this case, gave is used as a transitive verb. That's why we have to use the object pronoun me.
However, in casual speech, it's very common to hear people use object pronouns in places where they should be using subject ones. Although grammatically entirely incorrect, using me instead of I in this manner has become so commonplace that most people are not even aware of the fact that they're actually making a grammatical mistake. So, it's not unheard of to hear something like this in daily conversation:
Me and my wife are going to visit Paris this summer.
My advise to you: never use this kind of grammar in formal writing; as far as conversational English is concerned, it's entirely up to you to decide which grammar to use as long as you understand the consequences.
The former can be used as a subject and the latter as an object.
I was invited to dinner. They invited me to dinner.