the strategy to use very much depends on the required throughput. If you have just one bacterial strain that you want to test then you may spend more effort. If you want to test 100s or 1000s of bacterial strains, each one can only be touched by a single experiment. As a student I participated in the screening for new restriction enzymes. The following is a rough outline of this procedure.
For screening purposes we produced a lysate by sanitation and centrifuged it. We took an aliquot of it and incubate it with a DNA sample of known sequence. Probably the best choice here is the DNA of a bacteriophage such as Lambda (from dam- and dcm- bacteria). Incubate for 1 h at 37 degrees C and then run the sample on an EtBr agarose gel. If you see that the substrate DNA has fallen into fragments of defined sizes this means that a specific restriction enzyme was present in the bacterial extract.
Problems with this approach may be:
i) you do not know which buffer is best suited for your restriction activity. For screening purposes I would use two or three or four different buffers, e.g. those supplied with the enzymes from commercial sources such as NEB.
ii) all cells contain DNAses. This may make the banding pattern sometimes difficult to see. Try to play around with incubation time and temperature. There may be a situation where the restriction enzyme has higher activity or the DNAase has lower activity.
iii) sometimes you see all DNA having only low mobility (remains in the well) . This usually comes from proteins that bind to the DNA. A proteinase K digest will help.
In many cases you find enzymes with known specificities. You may generate a small database of banding patterns that result when Lambda is cut by any of the known enzymes and simply compare the bands you observe in the screening to these.