David, do you need to open or save in PSD?
In PSP, you can open PSD files in layers but it will not maintain the vector layers and those will be rasterized. Also, layer styles and such will not be maintained.
If you want to save in PSD, you can do so just like you would save any other file.
As for TIFF files, there is no chance in PSP. It does not open them with the layers and cannot save in that format.