I agree with David. I'm sure you would not be able to pull off your entire itinerary by train and bus with the limited time you have. Based on Internet research (my experiences last April may not be predictive of transportation options next month), these public-transportation options might exist for you:
Seville to Arcos: This is doable. You would take a train or bus to Jerez de la Frontera and then a bus to Arcos. Or you might possibly find a direct bus with a stop in Jerez but no transfer. It's not even a terribly long trip.
Arcos to Ronda: Although Rome2Rio.com says TG Comes buses run this route, I can't find a schedule and suspect this may be a high-season-only bus. A good bit of the transportation into Ronda is by train, and of course Arcos doesn't have a train station. That could result in a gap.
Zahara de la Sierra: In theory TG Comes buses go from Arcos to Algodonales, which is a 7-mile taxi ride from Zahara. Unfortunately, the taxi company mentioned on the Rome2Rio website has "Ronda" in its name; now that's a $cary thought. However, as with Arcos-Ronda, I cannot confirm current service to Algodonales online. Also in theory, TG Comes runs a bus from Zahara de la Sierra on to Ronda, but I can't find information about it online, either.
Grazalema: As further evidence that Rome2Rio.com is not to be trusted, it doesn't indicate the existence of any public transportation between Ronda and Grazalema, but I've found two Comes buses on that route. The first departs at 1 PM, which is sort of late for a day-trip. The bus takes 50 minutes. The return departs at 4:10, which would allow for only a very short visit. I think there are more buses later in the year; I made a day-trip from Ronda to Grazalema last April and don't remember having only 2 hours in Grazalema.
Ronda to Granada: There are a couple of 2-1/2 hour trains daily, departing at 1:15 PM and 5:14 PM. There are also buses, but you might have to change in Malaga, making the bus trip take much longer than the train. If interested in the bus options here, drill down on Rome2Rio.com to get links for the multiple bus companies involved.
As of April there was at least one company offering a one-day bus tour from Seville that included Ronda and I believe two of the smaller towns, one probably being Arcos. I opted not to do that, because I wanted more time in Ronda and planned to spend at least two nights there. If there had been a bus tour only to the small villages, I would have been happy to take that.
I think there are too many moving parts in your plan to make BlaBlaCar a good idea here. One driver who changed his mind could leave you stranded.
In all cases the schedules could be highly day-of-week sensitive. I checked a random weekday.