We want calculate $H^{k}_{S^1}(S^2)$. We can choose two open sets $U=S^{2} \setminus p_{+}$ and $V = S^{2} \setminus p_{-}$, where $p_{+}$ and $p_{-}$ are, north pole and south pole of $S^{2}$. There are fixed points of action and they are $S^{1}$-invariant.\ So $U \cap V$ is omotopic to $S^{1} \hookrightarrow S^{2}$ on which $S^{1}$ acts freely. So $$ H^{*}_{S^{1}}(U \cap V) \simeq H^{*}_{S^{1}}(S^{1}) \simeq H^{*}\left(\frac{S^{1}}{S^{1}} \right) \simeq H^{*}(pt) .$$ Moreover $$ H^{*}_{S^{1}}(U) \simeq H^{*}_{S^{1}}(p_{-}) \simeq H^{*}(BS^{1}) = \mathbb{R}[x_{-}] $$ where $x_{-}$ has degree $2$ end $H^{*}_{S^{1}}(V)= \mathbb{R}[x_{+}]$.\

So when $k>1$ $$ H^{*}_{S^{1}}(S^{2}) \simeq \mathbb{R}[x_{+}] \oplus \mathbb{R}[x_{+}] $$ and if k=1 or k=0?

We want to calculate $H^{k}_{S^1}(S^2)$. We can choose two open sets $U=S^{2} \setminus p_{+}$ and $V = S^{2} \setminus p_{-}$, where $p_{+}$ and $p_{-}$ are the north and south pole of $S^{2}$. They are fixed points of the action ($S^{1}$-invariant). 
So $U \cap V$ is homotopic to $S^{1} \hookrightarrow S^{2}$ on which $S^{1}$ acts freely. So $$ H^{*}_{S^{1}}(U \cap V) \simeq H^{*}_{S^{1}}(S^{1}) \simeq H^{*}\left(\frac{S^{1}}{S^{1}} \right) \simeq H^{*}(pt) .$$ Moreover $$ H^{*}_{S^{1}}(U) \simeq H^{*}_{S^{1}}(p_{-}) \simeq H^{*}(BS^{1}) = \mathbb{R}[x_{-}] $$ where $x_{-}$ has degree $2$ end $H^{*}_{S^{1}}(V)= \mathbb{R}[x_{+}]$.

So when $k>1$ $$ H^{*}_{S^{1}}(S^{2}) \simeq \mathbb{R}[x_{+}] \oplus \mathbb{R}[x_{+}] $$ and if $k=1$ or $k=0$?

We want calculate $H^{k}_{S^1}(S^2)$. We can choose two open sets $U=S^{2} \setminus p_{+}$ and $V = S^{2} \setminus p_{-}$, where $p_{+}$ and $p_{-}$ are, north pole and south pole of $S^{2}$. There are fixed points of action and they are $S^{1}$-invariant.\ So $U \cap V$ is omotopic to $S^{1} \hookrightarrow S^{2}$ on which $S^{1}$ acts freely. So $$ H^{*}_{S^{1}}(U \cap V) \simeq H^{*}_{S^{1}}(S^{1}) \simeq H^{*}\left(\frac{S^{1}}{S^{1}} \right) \simeq H^{*}(pt) $$. Moreover $$ H^{*}_{S^{1}}(U) \simeq H^{*}_{S^{1}}(p_{-}) \simeq H^{*}(BS^{1}) = \mathbb{R}[x_{-}] $$ where $x_{-}$ has degree $2$ end $H^{*}_{S^{1}}(V)= \mathbb{R}[x_{+}]$.\

So when $k>1$ $$ H^{*}_{S^{1}}(S^{2}) \simeq \mathbb{R}[x_{+}] \oplus \mathbb{R}[x_{+}] $$ and if k=1 or k=0?

We want calculate $H^{k}_{S^1}(S^2)$. We can choose two open sets $U=S^{2} \setminus p_{+}$ and $V = S^{2} \setminus p_{-}$, where $p_{+}$ and $p_{-}$ are, north pole and south pole of $S^{2}$. There are fixed points of action and they are $S^{1}$-invariant.\ So $U \cap V$ is omotopic to $S^{1} \hookrightarrow S^{2}$ on which $S^{1}$ acts freely. So $$ H^{*}_{S^{1}}(U \cap V) \simeq H^{*}_{S^{1}}(S^{1}) \simeq H^{*}\left(\frac{S^{1}}{S^{1}} \right) \simeq H^{*}(pt) .$$ Moreover $$ H^{*}_{S^{1}}(U) \simeq H^{*}_{S^{1}}(p_{-}) \simeq H^{*}(BS^{1}) = \mathbb{R}[x_{-}] $$ where $x_{-}$ has degree $2$ end $H^{*}_{S^{1}}(V)= \mathbb{R}[x_{+}]$.\

So when $k>1$ $$ H^{*}_{S^{1}}(S^{2}) \simeq \mathbb{R}[x_{+}] \oplus \mathbb{R}[x_{+}] $$ and if k=1 or k=0?