FCO to Positano....then Positano to Venice. ...first time to Italy

Getting from FCO to Positano: I recommend taking the Leonardo Express train from FCO to Rome's Termini train station (trains run every 30 min, trip takes 30 min, cost €14 per person, be sure to validate ticket before getting on train). From Termini, there are high-speed trains at least hourly to Salerno (2-2.5 hours, approx. €35 per person). See www.trenitalia.com for schedules and pricing.
From Salerno, you can take a boat, bus or taxi to Positano. Buses can be crowded and boats run on a limited schedule off season. The easiest option would be a taxi. There should be taxi's waiting outside the train station, Be sure to negotiate price before getting in. A recommended transfer company based in Praiano (near Positano) is Umberto & Giovanni Benvenuto. You could arrange for them to meet you at the Salerno train station and take you to your hotel in Positano - www.benvenutolimos.com/Transfer. They also do private tours of the coast and to nearby sites, as well as transfers. They could even pick you up at FCO, but this would be pretty expensive.

Another option would be to take the train to Naples instead of Salerno (high-speed trains hourly, takes 70 min., €34). From Naples Centrale station, you can go downstairs to the Garibaldi station and catch a commuter train to Sorrento called the Circumvesuviana (runs every half-hour, takes 70 min. to Sorrento, costs about €5). From Sorrento, you would then need to take a bus (cheap, frequent, but crowded) a boat or taxi to Positano. As with coming from Salerno, a taxi would be the easiest option.

Getting to Venice: You would reverse the above process to get back to Naples. From Naples, you can take a train (hourly, 5.5 hours, most with a train change in Rome) or fly cheaply to Venice if booked ahead. See www.skyscanner.com for options.

Once in Naples, not sure if flying is the fastest option.

If you take the train to Naples from Meta or Sorrento you arrive at the Napoli Central station.

From there you could take a train to Venice and get there in 5h 35min (change in Bologna in most cases). There are also 2 trains that get you there without train changes in 5h 05m. The morning one departs from Naples Centrale at 7:30 the afternoon departs at 16:30.

If you choose the flight option, first you need to get to the airport, then you need to go through check in and security (let's say 1.5 hours), plus upon arrival you need about 1.5 hours for deplaning, retrieve your luggage and travel from the airport to the city of Venice. So flying might save you an hour over the high speed train and the train is certainly more comfortable.
Cost-wise, once you include the cost of the transfers from/to the cities to/from their respective airports, you might not save much by flying, even with Easyjet. Easyjet is the only low cost airline that flies from NAP to VCE, so don't bother looking for anything else. The other airline is Meridiana and I don't think they can beat Easyjet's prices.

We are flying into Rome and going to meet friends for a week in Positano

This sounds attractive, but do you understand that it's ambitious? Positano is popular but out-of-the-way, and subject to road traffic and transit crowding. It's like a post here saying someone was staying North of London to visit family in Brighton (on the coast.) You have made a decision to make local travel a major part of your trip to Italy. As long as you go into this with your eyes open, it's fine. It would be helpful to know whether your budget is limited and what month of the year.

Take the bus or taxi to Sorrento. From Sorrento the Curreri Viaggi bus will take you directly to the Napoli airport. cost is 10 euro time 1H20m

http://www.curreriviaggi.it/inglese/E_index.html

From the Venice airport, take the ATVO Venezia Express bus to Piazzale Roma stop.
Purchase tickets in the arrivals hall. Don't forget to validate ticket in the machine.

From Piazzale Roma take the People Mover to the cruise port.

http://europeforvisitors.com/venice/articles/people-mover-to-marittima-cruise-terminals.htm.

Good info on Venice here.

You are all so helpful! I finally feel like this is doable. I am not a nervous traveler...At All! This is the first time I have felt uncomfortable about any travel connections but You are really giving me The Best advice! This trip is scheduled for 5/29/15 - 6/ 14/15.

Just an added thought. If you have not already booked your air, consider flying into Naples rather than Rome. You could then take the Curreri Viaggi bus to Sorrento and the ferry to Positano. Much more comfortable than the Circumvesuvian.

Also, the ferry from Positano to Sorrento for the trip to Venice The schedule would be the deciding factor
on this leg.

Bus schedule for Amalfi Coast.

http://www.sitasudtrasporti.it/public/tpl/campania/5070.pdf

Ferries.

http://www.coopsantandrea.com/en/servizidilinea/servizi.asp

Or, to get to Positano in May from Rome, you could always take a train to Salerno and then a ferry......

Now that you've given the dates, I can tell you that on Wednesday, May 28, 2014, around 3PM, the Circumvesuviana train to Sorrento was so crowded that we could barely get off the train, onto the platform at the end of the line, Sorrento. It was elbow to elbow on the platform. It then took over five minutes to reach the stairs and small elevator to street level.

During that week, whenever we saw the SITA bus to Positano (we never rode it), it had a line that was more than 49 (the average capacity of tour busses in the USA), which means those in line were going to have to wait for at least two busses.

We paid for an auto transfer back to Naples from Sorrento, which saved us sitting on our luggage on the Circumvesuviana, after paying for a cab in Sorrento to the station with our luggage. (It was about 100 Euros.) Although that was on a Monday at Noon, it took about the same time by car as it would have in the Circumvesuviana.

Note that there is only one, winding, road to Positano (no train), and it is one lane in each direction. Because budget wasn't too important for us, we hired a car and driver for eight hours to see, briefly, Positano, Amafli, and Ravello. It was over 200 Euros, but worth it to us. Your mileage may vary.

You've received lots of good advice so far. There are several transportation options, and in the same situation, I'd probably use something like this.....

• Fly into Naples rather than Rome. Take Curreri Viaggi Bus from the airport to Sorrento (cost was €10 PP last time I checked), and then either Bus, Taxi or car service from there to Positano. You're going to be tired and jet lagged, so this is probably the easiest method.
• For the trip from Positano to Venice, either take the Bus from Positano to Sorrento, Circumvesuviana to Napoli Centrale and then Freccia to Venice OR if your budget will allow, arrange a car service from Positano to Napoli Centrale and then take the Freccia to Venice. Which option to choose will depend on your budget.

A few other points to note......

• From Napoli Centrale, you can either use Trenitalia or Italo for the trip to Venice. If you're willing to commit to travel at a specific time, you can save some money by pre-purchasing tickets. Note that the tickets will include the compulsory reservation and can ONLY be used on the one train, date and departure time specified on the ticket. On the trains with compulsory reservations, you'll be assigned a Car (Carrozza) number and Seat (Posti) number.
• As mentioned in the first reply, if you'll be using any Regionale trains such as the Leonardo Express from FCO to Roma Termini, you MUST validate the ticket prior to boarding the train on the day of travel, or you'll face hefty fines which will be collected on the spot! The validation machines are either bright yellow or green & gray, and will be located in several places around each station.
• Regarding private transfer services for getting back and forth from Positano, there are reliable firms listed in the guidebook.

Ken is back in action! Welcome back Ken.

Fear mongering is a little strong James. Do you recommend that people fly to Washington, DC for a visit to New York City? That's the equivalent. At least in Washington, you can get a single Bolt Bus to New York City, instead of a three-seat (or more ... ) ride through the worst-organized part of Italy.

Of course, you couldn't tell from my posts, but WE combined the Chelsea Flower Show with Sorrento earlier this year! That's much more ambitious than the OP. But we made the decision based on low-fare airlines from suburban London to Naples, not to Rome. And we packed our bags to match the airlines nasty limitations. I'd hate to take a cruise with 20 pounds of luggage. I'll take the heat for our decision!

If my posts showed fervor (not "fear"), it's because so many people posting here plan trips to Positano and then discover (for example) how hard it is to see Pompeii from Positano. They apply American standards of daytrip travel to a very distant location. Note that I haven't stated my opinion of Positano, because, it wasn't asked for.