Isolating the gene of interest for protein production
As you already wrote there are two basic ideas: Synthesis of the desired gene sequence and use of existing DNA.

DNA Synthesis is getting cheaper so it is becoming the method of choice for generating DNA-samples up to about 5000 bp (for example see the IDT-page for commercial gene synthesis). If this is too short for your desired sequence there is the possibility to split the gene to different of those synthetic sequences and to fuse them using Gibson-Cloning (Gibson et al. 2009) - after adjusting the synthetic sequences so that there is an overlapping sequence. Alternatively you could use classical cloning - provided there are suitable cleaving sites in your sequence.

If you don't want to synthesize the DNA and have a template available you could design suitable primers and amplify the gene from the template using PCR and purify it by gel-electrophoresis. Inserting the sequence into a vector then requires that you find suitable cleaving sites within the sequence and the vector.