There are lots of questions when doing purification like what buffer to choose, salt concentration, additives, inhibitors, temperature, purification steps likewise. Use of homogenizer or sonicator is trivial and depends on the scale and purpose of the purification.
It very much depends upon what you intend to use the protein for. If you want native protein for purification or enzyme measurements, then you need a fairly gentle protocol. Traditionally bacterial cells are broken with a French press or a sonicator, depending upon the volume of cells. French press is good for larger quantities and sonicator best for smaller. You can also lyse the cells by freeze thaw cycles after lysozyme treatment. Please note that these methods generally do NOT release membrane bound proteins, those need to be released using detergent. On the other hand, if you want total protein to run on a gel or some other procedure but don't require that the proteins be fully folded and native, then lysis by detergent is the most efficient method.
I could be wrong, but I thought homogenizers are good for larger cells but not very effective on smaller bacterial cells.
I used lysis buffer containing Triton X-100 and PMSF. Also, I used two different ways to break up the bacterial cell wall which are homogenisator and sonicator.