Who is first?

Since B and D contradict each other, exactly one of them must be true.

Since A and H agree with each other while E and F are against them, exactly two of these four must be true.

Therefore, neither C nor G guessed correctly. Since G guessed that C is not first, $\boxed{C}$ must be first.

Let us go through all the guesses. If A's guess is correct or true, then either F or H is first. Then, B's guess that he/she is first is false. C's guess is false, and so on. The resulting truth table if we assume A's guess is true is as follows.

\[\begin{array} {} \text{A:} & \text{Either F or H is first.} & \color{blue} \text{True} \\ \text{B:} & \text{I'm first.} & \color{red} \text{False} \\ \text{C:} & \text{G is first.} & \color{red} \text{False} \\ \text{D:} & \text{B is not first.} & \color{blue} \text{True} \\ \text{E:} & \text{A is wrong.} & \color{red} \text{False} \\ \text{F:} & \text{Neither I nor H is the first.} & \color{red} \text{False} \\ \text{G:} & \text{C is not first.} & \color{blue} \text{True} \\ \text{H:} & \text{I agree with A.} & \color{blue} \text{True} \end{array} \]

Since the truth table above shows that four and not three persons guessed correctly, A's guess is false. Now, assume that B's guess is true, then the truth table is as follows:

\[\begin{array} {} \text{A:} & \text{Either F or H is first.} & \color{red} \text{False} \\ \text{B:} & \text{I'm first.} & \color{blue} \text{True} \\ \text{C:} & \text{G is first.} & \color{red} \text{False} \\ \text{D:} & \text{B is not first.} & \color{red} \text{False} \\ \text{E:} & \text{A is wrong.} & \color{blue} \text{True} \\ \text{F:} & \text{Neither I nor H is the first.} & \color{blue} \text{True} \\ \text{G:} & \text{C is not first.} & \color{blue} \text{True} \\ \text{H:} & \text{I agree with A.} & \color{red} \text{False} \end{array} \]

Again, there are four correct guesses, hence B's guess is false. Now, assume that C's guess is true.

\[\begin{array} {} \text{A:} & \text{Either F or H is first.} & \color{red} \text{False} \\ \text{B:} & \text{I'm first.} & \color{red} \text{False} \\ \text{C:} & \text{G is first.} & \color{blue} \text{True} \\ \text{D:} & \text{B is not first.} & \color{blue} \text{True} \\ \text{E:} & \text{A is wrong.} & \color{blue} \text{True} \\ \text{F:} & \text{Neither I nor H is the first.} & \color{blue} \text{True} \\ \text{G:} & \text{C is not first.} & \color{blue} \text{True} \\ \text{H:} & \text{I agree with A.} & \color{red} \text{False} \end{array} \]

There are five correct guesses and C's guess is false.

\[\begin{array} {} \text{A:} & \text{Either F or H is first.} & \color{red} \text{False} \\ \text{B:} & \text{I'm first.} & \color{red} \text{False} \\ \text{C:} & \text{G is first.} & \color{red} \text{False} \\ \text{D:} & \text{B is not first.} & \color{blue} \text{True} \\ \text{E:} & \text{A is wrong.} & \color{blue} \text{True} \\ \text{F:} & \text{Neither I nor H is the first.} & \color{blue} \text{True} \\ \text{G:} & \text{C is not first.} & \color{red} \text{False} \\ \text{H:} & \text{I agree with A.} & \color{red} \text{False} \end{array} \]

As we go through the guesses, if we assume G's guess is true, then there were four correct guesses. If we assume G's guess is false, there were exactly three correct guesses. Hence G's guess that "C is not first." must be wrong. That means C got first.