Almalfi Coast 2 day trip advice needed

Not enough detail to advise.
Where are you coming from? Where are you staying? Can you really only allocate two days? It is woefully brief, so you need to prioritize.
You could possible stop at Pompeii en route, then enjoy evening in one of the villages. The next day, enjoy some other villages by ferry or bus.
Sorrento is expendable from this list, and I would probably leave out Minori and Praiano.

Yes, more information would be helpful. How many NIGHTS are you staying on the coast and where? Where are you coming from on arrival day?

Basing in Rome but delaying the Rome portion by 2 days to spend 2 days on the Amalfi coast.
Taking a train from venice to Rome. From Rome termini we are staying in Sorrento for a night's stay. The next day we spend the whole day on the Amalfi coast(various stops including Positano, Priano, Amalfi, Minori and staying the night in Minori before going back to Rome the next day.
Trying to figure out if it is best and the easiest to drive the whole time with a rental car or if we can do regional trains, ferries, and the sita bus line to get all of the amalfi coast done. Also trying to figure out if the private driver is a good option as well and how much those usually run.

If you insist on seeing all of those in one day, I would look into the private driver, which is the only conceivable way it could occur.
You do not need to split the two nights. This pace will only check boxes and not allow you to enjoy the AC, and I urge you to consider scaling back. These are tiny villages meant to be savored.

You don't have to stop in Rome when you go to Naples. Do instead.

Venice-Bologna-Naples. You change trains in Bologna which is a smaller station.

Edit. There are 2 direct trains at 7:25 and 10:25 that take a little over 5 hrs.

Use this to look up train schedules.

http://www.trenitalia.com/tcom-en

this appears to be the most difficult portion of the trip to navigate.

Adam, it's "difficult" because you're trying to do too many places in too short a time. You also don't realistically have "2 days" for the coast: you have 1 full day and whatever little time you will have left after getting there from Venice. Any way you work it, that's going to involve a big chunk of the day.

I might look at high-speed train to Naples, and then...
A. Change to local Circumvesuviana train to Sorrento.

B. Take the Campania Express train to Sorrento if your trip is before Oct 14th and the timing works:
http://www.napoliunplugged.com/campania-express-tourist-train-to-herculaneum-pompeii-and-sorrento

C. Transfer to the docks in Naples by taxi and take a ferry to Sorrento
https://www.sorrentoinsider.com/en/ferry-schedule?from=napoli&to=sorrento#

The next day, maybe take a ferry to Amalfi and work your way back to Sorrento on SITA buses. You might be able to do both Praiano and Positano but that's more than I'd try to squeeze into a day. Minori is east of Amalfi and would involve backtracking via bus from Amalfi so I'd drop that one. No, I wouldn't recommend changing hotels for just one night either: you'd have to drag your luggage around all day if using public transport.

Then get on the Circumvesuviana or Campania Express the next day, do Pompeii enroute to Naples, and then take high-speed train from Naples back to Rome. I'm not a fan of private drivers (my stomach can't take the backseat of a car, and especially not on winding Amalfi/Sorrentine Coast roads. They also do not come inexpensively) but I'm sure someone else will offer up some info there.

Your OP raises a host of questions, so many that I suggest you use the Search box (note the filters offered after you Search) to read the many posts we get about this area, and then posts about renting cars in Italy for any purpose. The Amalfi Coast is not "remote" but it is not that easy to get to. I personally would not prioritize it for such a short visit, especially since this is your first trip to Italy. How did you select Amalfi, as opposed to Positano or some other town? Do you understand that Sorrento is not, technically, "on" the Amalfi Coast, or that Sorrento doesn't really have beaches? Do you know that the small beaches on the A.C. are all pebbles, and the water tends to be cold?

Some specific items: About four years ago, we paid 90 Euros for a one-way transfer from our Sorrento hotel to the Naples rail station. We paid 240 Euros for a round-trip A/C Mercedes sedan and driver (not a skilled guide, but knowledgeable about his neighborhood ... ) for 8 hours, seeing Positano, Amalfi, Ravello from the hotel in Sorrento. We're not afraid of curvy mountain roads, we wanted to avoid summer traffic and impossible parking situations and ZTLs.

October may be better, but this area books heavily, six to twelve months out. You are unlikely to get your first choice hotel, even in Amalfi the town, in shoulder-season October. I wonder if the touristy shops may have some closures by then, in, say, Positano. Sorrento is almost year-round, for sun-seeking UK residents, for example.

Edit: Even off-season, it is unlikely that you can visit four Amalfi Coast towns by PUBLIC TRANSPORTATION in one day. They just aren't that well-connected. Ferries run less often in the fall, and are affected by daily weather, and have less frequency than busses, anyway.

If you really want a 'coastal' experience on this trip, I think life would be easier (and more enjoyable) if you were to substitute Cinque Terre for the AC. Much easier to get to CT with your Venice to Rome routing. You could spend both nights in one place in CT and actually take time to enjoy being there vs your planned 'drive by' of the AC

I agree with Kathy that you are trying to do too much in too little time

That is my $0.02