IMO, out of the 3 types of tests auDNA, yDNA, and mtDNA, the best test you should take is autosomal for yourself.
Given you are looking for your father's history, I presume you are interested in all lines, including your grandmothers. auDNA is the only real option for this.
All DNA tests are useful to some degree, but if you get a male relative agree to a yDNA test, you would still need to or at least should confirm your DNA connection to that person.
auDNA is like a season ticket to the lottery. You can't win unless you purchase a ticket. There are new subscribers every week.
It is true that you will probably find many relatives that might not be that close, but who knows, you might get lucky and find someone who is close enough. auDNA has been successful in many cases of adoptees finding parents, so I don't see why finding great grandparents is much different.
When choosing an auDNA test, I might go for AncesteryDNA. They will probably be the fastest growing database where you live, and also, based on what I have been reading, has the best turnaround time. It will give you some idea of what to expect. You will have to upload your raw DNA to www.gedmatch.com and/or transfer the Raw Data to FTDNA to get the maximum benefit.
You don't mention your mother. If testing her is a possibility, I would order a kit for her or another willing relative.