Geodesic flows and Curvature

"For instance given a point $p$ on $M$, if $p$ can be connected to any other point on the manifold by a geodesic (as in sphere) then does the manifold have constant curvature?" I don't think it's hard to construct counterexamples to this. Take a point $p$ in the Euclidean plane. Now pick some region $R$ that doesn't include $p$, and introduce some small change in the metric $g\rightarrow g+\delta g$ that only occurs within $R$, so that the Gaussian curvature no longer vanishes inside $R$. From $p$, you can send out a geodesic at any angle $\theta$. As you increase $\theta$, these geodesics sweep the plane like the beam of a searchlight. It seems pretty clear to me that if $\delta g$ is small, then we will still cover the entire plane with these geodesics.

Some clarifications:

Every complete Riemannian manifold has the property that for any two points $p, q \in M$, there exists a geodesic $\sigma_{pq}$ connecting $p$ and $q$. Moreover, the geodesic can be chosen to be minimizing, that is, there are no other curves $\alpha : I \rightarrow M$, geodesics or not, with length strictly less than the length of $\sigma_{pq}$. This is part of a basic result known as Hopf-Rinow's Theorem.

Thus, if it were true that being able to connect any $p \in M$ with any other $q \in M$ by a geodesic implies constant curvature, then every complete Riemannian manifold would have constant curvature, which is obviously false.

A related property which does imply constant curvature is homogeneity. A Riemannian manifold $(M, g)$ is homogeneous provided that for any $p, q \in M$, there exists an isometry $\Phi_{pq} : M \rightarrow M$ sending $p$ to $q$. The intuition behind this is that the metric looks the same at every point, and thus everything metric-related (like curvature) must be constant.

About the isometry group: the result is that if $(M, g)$ is Riemannian (finite-dimensional, I'm assuming), the its isometry group is a Lie group. There are no additional conditions. A lot is known about isometry groups of Riemannian manifolds, and since you like Kobayashi, you can take a look at another one of his books, called Transformation groups in Riemannian geometry, which has a nice exposition about this topic and many others.