A geodesic in a plane $P$ writes as $\gamma(s)=p+s v$, $p\in P$ and $v\in T_pP=P$. Then the exponential map is simply $$exp(v)=p+v$$
that is, the exponential map at $p$ is a translation on $P$ with translation vector $p$. Let us observe that the $\mbox{exp}$ is a diffeomorphism in the whole $T_pP$ and thus the normal neighborhood of $p$ is the very plane $P$.
In the unit sphere ${\mathbb S}^2$ a geodesic is $\gamma(s)=\cos(s)p+\sin(s) v/|v|$, where $p\in {\mathbb S}^2$ and $v\in T_p{\mathbb S}^2$. Then $$exp(v)=\cos(1)p+\sin(1)v/|v|.$$ We want to study when $\mbox{exp}$ is one-to-one. By the property $\gamma(t;p,\lambda v)=\gamma(\lambda t; p,v)$, we have $\mbox{exp}(\lambda v)=\gamma(t)$. If $|v|=1$, for $\lambda=\pi$, $\gamma(\pi;p,\lambda v)=-p$. This implies that the exponential map $\mbox{exp}$ is one-to-one in the ball $B_\pi(0)$ of radius $\pi$.
