Should I use spot color or process color to color the vector images in AI to work best with four color printing?
You're printing in process so use process colors.
Don't use any spot colors (e.g. Pantone colors). They are meant for printing with specific singular inks, so will need to be converted back to process colors if you're printing exclusively in CMYK; giving you inconsistent and unexpected color changes.
What is the best way to place the AI vector images into InDesign? The current method I am using is exporting the vectors as PDF files and then "Place" them in InDesign. Is this ok? Does exporting vector images as PDF alter the images' colors at all?
That can give you problems, but it completely depends on how you save your PDFs.
You can, when saving a PDF from Illustrator, choose to convert your colors to a certain color profile, converting the color values to that new profile and embedding that profile. You can also leave the colors unmanaged and save the PDF with full Illustrator editing capabilities; essentially leaving you with an Illustrator file with a PDF extension.
If you do save your file as a PDF to place in InDesign, save it from Illustrator using the [Illustrator Default] preset, which won't do any color conversion or embed any color profiles.
To save any uncertainty though, the best option is to simply place your Illustrator document (*.ai) directly in InDesign. This saves you a step in your process and cuts down on unnecessary extra files to manage. Make sure you're working in the same intended color space in Illustrator and InDesign and place your working files directly and you shouldn't have any problems.