Train tickets

No need for first class on a one hour trip like that. The regular coach car is just fine. You'll save some money buying that ticket in advance and if you know exactly which train you want, I would purchase in advance.

Yes you will get the best price by buying a nonrefundable ticket now. The pricing comes in 3 categories, Super Economy, Economy and Base, with increasing prices and greater flexibility. Looks like the train you want has 6 classes of service, Standard, Premium, Business Salottino, Business Silenzio, Business, Executive. You can click on the little information icon to see what you get. Standard and Premium are 2nd class seats with Premium having nicer fabric and a few perks, Business are different styles of 1st class, and Executive is super 1st class.
http://www.trenitalia.com/tcom-en

...I know this has been asked a million times.... But.... Again.... You are correct. But did you read the answers? Because the answers are nearly always the same. The ONLY reason (but it is a good one) to buy on line is to take advantage of any available discount. Super Economy tickets can be very cheap. Also SE tickets are no refund, no change no matter how good your story is. Economy allows one change paying the difference in fare if applicable. And Base is the standard walk up fare that can be exchange anytime prior to the station leaving the station. The one exception is Regionale Trains (R). There is no discount and should be purchased only in the station at a ticket window, ticket machine, or from a travel agent. These tickets much be validated (time stamped) prior to using but you can use them on any R train on any day at time.

Use trenitalia.com. The web in English is easy to use and understand.

Does it make sense to buy Trenitalia tickets in advance?

Frank did an excellent job answering that: there can be an advantage IF you can land super economy fares and IF you are sure you can commit to that specific train/time. Economy fares can also be worthwhile, and can be changed once for a difference in price if you miss your train (see the Trenitalia website for details.)

I have a pretty tight schedule from Rome to Positano ( with a stop in
Pompeii and Sorrento) and there is only one option that will get me
there in time and it's a 7:35 AM Rome to Naples (high speed/Termini)
which I would imagine will be quite crowded (Friday). Should I go
ahead and buy ticket?

That is a LOT to try and do in one day but see the above: it all depends on if you want to save some $$ by doing so, and if super-economy/economy tickets are available.

If I want to go First Class...how do I distinguish?

Personally, I don't see a need to book first class (we never do.) It's not that long a journey from Rome to Naples, and not that big a difference between 1st and 2nd class to warrant a big price difference. Buy the cheapest fare for the time you want to depart Rome on a direct, high-speed train. 1st and 2nd-class carriages have a 1 or 2 on the side plus a number that will correspond to one on your ticket.

Lastly, you do understand that you cannot get to Pompeii or Sorrento on a Trenitalia train, or Positano on a train at all?

Are you trying to make the train connection directly after an inbound international flight? If so, buying advance tickets is not advisable.

Buying the more expensive first class tickets won't get you there any quicker. Both classes of coaches will be traveling at precisely the same speed. It won't matter how crowded the train is, as all passengers will have compulsory seat reservations. If you get a ticket, you have a seat.

You can also buy tickets at www.captaintrain.com, which is a more user friendly website.

It would help to have a bit more information. What time do you have to be in Positano? How long are you planning to stop in Pompeii and Sorrento? My impression so far is that you may be trying to do too much with the time you have available.