I want to take a series of photos of a still object from changing locations. All photos should have the same aperture, shutter and color balance to avoid any differences in brightness/color in the sequence. The focus can change. Ideally, I want the camera to figure out the exposure for the first photo. My thinking: take one picture in an auto-mode, write down the EXIF values, switch to M mode, set values to the ones written down, and shoot away. Any better ideas? The camera is Sony NEX-5R, if it matters.
Seems like you're just asking about how to generally meter a scene, with no special lighting conditions. There are many metering questions you may find helpful.
Using the camera's built-in meter as you describe would be fine. You don't need to use auto mode though. Just switch to spot metering in manual mode and look at the meter indicator as you adjust settings. The relevant settings are aperture, shutter speed, ISO. People usually know 2/3 of the settings they want, and use the meter to figure out the last one. You can also use an external incident-light meter for the basic scenario you describe.
I suggest using spot metering because you mention being particularly interested in exposing for a specific object across multiple photos. Other metering modes measure exposure for entire scenes.
I would recommend against relying on the visual appearance of the scene in the LCD because the LCD is deceptive. Blown-out highlights are often not visible on the LCD, unless you have "zebras" enabled.
Histograms also contain information about exposure, along with some additional information. As with non-spot metering modes, histograms reflect the exposure of the entire scene.