We report the first measurements of transverse single-spin asymmetries for inclusive jet and jet + pi(+/-) production at midrapidity from transversely polarized proton-proton collisions at root s = 500 GeV. The data were collected in 2011 with the STAR detector sampled from 23 pb(-1) integrated luminosity with an average beam polarization of 53%. Asymmetries are reported for jets with transverse momenta 6 < p(T) < 55 GeV/c and pseudorapidity vertical bar eta vertical bar < 1. Presented are measurements of the inclusive-jet azimuthal transverse single-spin asymmetry, sensitive to twist-3 initial-state quark-gluon correlators; the Collins asymmetry, sensitive to quark transversity coupled to the polarized Collins fragmentation function; and the first measurement of the "Collins-like" asymmetry, sensitive to linearly polarized gluons. Within the present statistical precision, inclusive-jet and Collins-like asymmetries are small, with the latter allowing the first experimental constraints on gluon linear polarization in a polarized proton. At higher values of jet transverse momenta, we observe the first nonzero Collins asymmetries in polarized-proton collisions, with a statistical significance of greater than 5 sigma. The results span a range of x similar to results from semi-inclusive deep-inelastic scattering but at much higher Q(2). The Collins results enable tests of universality and factorization breaking in the transverse momentum-dependent formulation of perturbative quantum chromodynamics.