Best way to travel from Rome to Palermo

It's a long way via ground transportation (trains). Like 12 hours. Flying is the best way to go (1 hour). But if you already have a eurail pass you might as well use it, I guess. You can see train schedules on http://www.trenitalia.com.
(STATION NAMES: from: ROMA TERMINI to: PALERMO CENTRALE)

For Sicily planning check this website:
http://www.bestofsicily.com/

I think you need to define, "best". For me, it would be to get a cheap flight; but then, I wouldn't dream of going to Sicily in August. It's likely to be really HOT! If you go, be sure to get lodging with AC.
Is time or money more important to you?

Much of Italy will be hot in August, not just Sicily. Be prepared.

If you do want to investigate flying, note that Sicily has four airports. Catania is actually the biggest with the most flight options, followed by Palermo. Trapani is smaller, and Comiso is very small with few flights. As for the mainland, Rome has by far the largest selection of flights to the Sicily airports, followed by Naples and Milano Linate. Other airports on the mainland usually have 1-2 flights a day to Catania and Palermo.

To investigate flights, use Skyscanner: http://www.skyscanner.com/. Note that Alitalia can be cheaper than the budget carriers when you add in all the extras. For example, Alitalia includes one carry-on, while most others charge for this. Alitalia's International website has better prices for domestic Italian flights than their US site, and unlike their Italian website, it has an English option.

A great itinerary is to start in Palermo and make your way to Catania or vice versa. Even if you're just hitting some highlights, you need 10-14 nights. If you have less time than this, it's probably not worth the time and hassle of getting there and back.

Another way to get to Sicily is the overnight ferry between Naples and Palermo.

If you don't mind a long read and are looking for ideas, my VERY detailed Sicily trip report is here: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina

I agree with Harold: You need time in Sicily to make the hassle of getting there worthwhile. And it will be mega-hot. But it is fabulous.

I took an overnight train from Rome to Catania last June. I was in a second-class cuchetta--the cheapest form of horizontal accommodation. I used to be able to get some sleep doing that, but last year's trip was dreadful because of rough tracks that had the train carriage constantly jerking back and forth. I know how tempting it is to take night trains when you have a rail pass, but I do not recommend this one. There was also the matter of a 3-hour delay at the Straits of Messina due to track work, which may or may not still be going on.

All of Italy will be hot in August, not just Sicily. However, Sicily is an island, and therefore kissed by nice breezes near the coast. It's not that bad actually near the water. At high altitudes, such as on top the famous volcano Mt. Etna, it is cool as well.

The interior valleys however can be brutal (worse than Walnut Creek or Antioch or even Stockton), but so is Bolzano in Italy's far north next to the mountains.

We are looking at a multiday road trip from Rome to Palermo with the travel being a part of the experience. We are looking at it being a 3-4 day trip with the travel part of the experience. We want to travel along the cost. Any suggestions for little towns to stay in along the way?

Since you will be traveling by train throughout your trip, I would suggest getting to Palermo as efficiently as possible....by plane. Taking the "Leonardo Express" train from Rome Termini to Rome airport is very efficient. We flew to Palermo, then took a reasonably priced shuttle bus from the airport to the city. The train ride from Rome to Palermo is long and tedious. I'd rather spend the time exploring Palermo. We really enjoyed Palermo. It is culturally diverse, with a wide range of neighborhoods...both modern upscale and more traditional old world. Be sure to visit one of the major markets....very colorful and a good way to be among the locals. Depending on where you are traveling in Sicily, you might choose to return to the mainland by ferry or from the airport in Catania. Have fun!

You could stay in Paestum and Tropea, but in 3-4 days you will be turning around, having seen almost nothing of Sicilia, unless your road trip is time on top of your stay in Sicilia.

I am reading into your question that money may be an issue. If so use the rail pass and see select sights on your way. You can sometimes find really cheap airfare from FCO to Catania. I think mine cost me $75 one way. Sure saves the time.