Where do the lipids and proteins constituting the cell membrane get synthesised?

Lipids and proteins are synthesized in the ER [Endoplasmic Reticulum].

Lipids are synthesized by smooth endoplasmic reticulum which is also called as SER.
Proteins are synthesized by rough endoplasmic reticulum which is also called as RER. Proteins are synthesized by the ribosomes that are attached to the RER.