Production and state-selective detection of ultracold molecular rubidium (⁸⁵Rb₂) in the singlet ground state and the metastable triplet state are described. Ultracold rubidium molecules are produced via photoassociation followed by radiative decay. State-selective detection of high-v ground-state and metastable- state molecules is carried out by resonant two-photon ionization (R2PI) with a pulsed laser. The spectra of both the ground singlet and metastable triplet states are investigated. Spectroscopy of several intermediate excited states, some of which are observed for the first time, is also studied. Mixing between excited singlet and triplet states is observed in both the singlet and triplet spectra. A proposal for producing v=0 ground-state molecules by utilizing this singlet-triplet mixing is given. Magnetic trapping and filtering of metastable triplet-state molecules is also described, and a scheme for optical trapping of molecules is proposed.