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There are several types of bordism. If we are talking about unoriented, oriented, or complex bordism, usually denoted MO, MSO and MU respectively, then the first statement (about the bordism groups of a point being known) is true. However if we are talking about framed bordism or symplectic bordism,
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bordism groups MO_*(M) of a space M are known once you know the mod 2 cohomology groups of the space (which are computable if M is homotopy equivalent to CW complex, eg a manifold).
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In what sense are higher dimensional manifold "classified by surgery theory"? This is mentioned several times and is potentially misleading.
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the bordism groups of a point are not known, eg in the framed case they are the stable homotopy groups of spheres.
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