How many equivalence classes in a set of well-orders of a set

Question

The question: Let $S$ be a set and $\operatorname{wo}(S)=\{X: X\subseteq S \land (X,\le) \text{ is a well order}\}$. Furthermore, partition $\operatorname{wo}(S)$ into equvialence classes based on order-isomorphic. Call these classes $W_1,W_2,\ldots,W_\omega,\ldots$ and let $W=\{W_1,W_2,\ldots,W_\omega,\ldots\}$.

1. How many elements in $W$ are there?

2. Prove that $|W|>|S|$

My thoughts: Each equivalence class $W_i$ will be order-isomorphic to an ordinal number. I am thinking that this means that $|W|=|S|+1$ as the elements in $W$ will be order-isomorphic to $0,1,\ldots,S$ respectively. Could anyone please help me prove this?

For 2., I am a bit unsure. If $|S|=\omega$, say, then $S=\omega$. I'm not sure then whether we should view $|W|$ as $\omega+1$ or $1+\omega=\omega$. Thus, I am not sure how to prove this assertion.

Any help would be greatly appreciated :)