I think you are almost right. DNA (in the nucleus) is transcribed to mRNA, and protein synthesis occurs on ribosomes where the the sequence of bases in the mRNA is translated into protein. Ribosomes are located in the cytoplasm or attached to the endoplasmic reticulum. (The 'mRNA story' is well told in Who discovered mRNA?)
But there is also another type of RNA that is critical to protein synthesis, and that is transfer RNA (tRNA): each amino acid is attached to an 'adaptor' molecule (tRNA) and it is this form of the amino acid that is used by the ribosome. It is the adaptor molecule (a tRNA 'charged' with an amino acid) that contains the anti-codon, which base-pairs with the mRNA during protein biosynthesis.
In order to carry out protein biosynthesis, the cell needs a supply of amino acids, the 'building blocks' of proteins. It is here your thinking (maybe) is a little confused? The building blocks may come the digestion of protein in food, or they may be synthesized by the organism. If they can be synthesized 'de novo', perhaps starting from a glycolytic or Krebs cycle intermediate, they are (usually) considered non-essential. What amino acids are considered essential or non-essential depends on the organism. Humans, for example cannot synthesize 'de novo' the aromatic amino acids Phe, Tyr or Trp, but we can convert Phe to Tyr. E.coli, on the other hand, can make all three of these amino acids from simple building blocks, but cannot convert Phe to Tyr (see Miller & Simmonds). Thus Tyr is considered non-essential for humans (provided that an adequate supply of Phe is available).
The biochemical pathways leading to amino-acid biosynthesis are usually quite complex, with the involvement of many gene products (enzymes). The story of amino acid biosynthesis is well told in Umbarger(1978):Amino Acid Biosynthesis and its Regulation.