pET21 is a rather classisal pET vector, which means that it is the T7 promoter that drives the expression of your gene of interest. You need an appropriate strain (like BL21) that harbours the T7 RNA polymerase gene to achieve any reasonable expression. In addition, you have to induce the T7 RNA polymerase gene inserted into the chromosome, as well as release the Lac repressor protein (LacI) from the operator region (located between the promoter and RBS on your vector). Induction is normally achieved by addition of IPTG rather than lactose. IPTG when added is not metabolized, so you expect that its concentration remains stable throughout the experiment. On the contrary, lactose - although in principle it could bind to the repressor protein after conversion to allolactose - will be degraded by the chromosome-encoded LacZ enzyme. As a consequence, the concentration will drop in the course of the experiment.
I never really used lactose to induce expression with pET vectors, although 'promoter leakage' that we sometimes observe in our experiments is most likely due to residual lactose present in the growth medium. To my opinion, 1 mM lactose should be sufficent to start the induction, but I would advise you to eventually add some more if your induction experiment exceeds 4 hours.
In my experience if you want to use lactose, nothing beats Studier's autoinducing media (described at PMID 15915565, but there are commercial alternatives available on the market. I am partial to ZYM5052). The only limitation vs. IPTG is that you must use lacY+ strains, which automatically eliminates most of the 'normal' cloning strains.