We carried out an extensive RXTE campaign, in 1997 June, to study Circinus X-1 during the active portion of its 16.55-d intensity cycle. The observations spanned 10 days, including 56% coverage for 7 d, and allowed us to find time segments which clearly demonstrate continuous evolution along the horizontal, normal, and flaring branches (HB/NB/FB) of a Z-source low-mass X-ray binary. These results confirm and extend the behavior we inferred from earlier observations. Here we study the continuous evolution of the Fourier power spectra and the energy spectra around the complete hardness-intensity track. A narrow quasi-periodic oscillation (QPO) peak, previously observed in the power spectra at 1.3-32 Hz, increases in frequency from 12 Hz to 25 Hz moving down a vertical extension of the HB in the hardness-intensity diagram. These horizontal branch QPOs (HBOs) occur near 30 Hz and fade in strength on the horizontal portion of the HB, while a broad peak in the power spectrum arises near 4 Hz. This peak becomes much more prominent along the NB and remains near 4 Hz (the normal branch QPOs, or NBOs). On the FB, neither QPO is present and the power spectrum is dominated by very low frequency noise. We also found that each branch of the spectral track is associated with a specific type of evolution of the energy spectrum. We explored various models for the energy spectrum and parameterized the evolution of the spectrum in terms of a two-component model consisting of a multi-temperature disk blackbody and a higher-temperature (~2 keV) blackbody. We also show that an unusual line- or edge-like feature occurs at ~10 keV in energy spectra from the FB and lower NB. This unusual feature is very similar to one seen on the FB and lower NB of the Z source GX 5-1.