I will be traveling from Rome to Capri in June and was looking at the train schedules and costs. Do I need to buy tickets before arriving in Rome or can I buy tickets the day of the trip?

You can usually buy tickets the day of the travel. You just pay more than twice the price you would pay now.
Just look at the last minutes vs. advance fare that you have been looking at, such as at Acquista il biglietto con le nostre offerte - Trenitalia I picked dates and time out of hat to compare the Rome-Napoli segment.

Roma Termini-Napoli Centrale 09:20-10:32 Frecciarossa 9605. If purchased now, the prices on two different travel dates:
March 13 = 45.00EU
June 5 = 16.90EU

We often purchase tickets last minute to allow us more flexibility while traveling. One disadvantage of purchasing at the station on the day of travel is that ticket lines can be slow in Italy. There are ticket machines in larger stations such as Rome, but when you take the inevitable pause to consider such things as fast train vs. local train, watch out for people offering to assist you. They can be persistent and push buttons for you and then press you for large tips. The alternative is to go online, once you are ready to book, and purchase tickets. You can easily access your ticket on your phone and even avoid any need to validate the ticket. Simply show the ticket taker your phone on board.

Check www.seat61.com for all you need to know about buying Italian train tickets - for general info trains BETS-European Rail Experts and www.ricksteves.com - like difference between 1st and 2nd class - first class tickets also discounted and at times not much more.

Most train stations in Italy now have ticket machines, but smaller stations often have limited opening hours. The ticket machines are multi-lingual and easy to use.

You can also buy tickets at a travel agent in the larger stations. They may charge a small commission. The ticket agents often have long queues in the larger stations.

do ticket machines accept American credit cards? Cash I presume?

If 2nd class by some chance sold out first class rarely would be.

If you're almost certain of the date and time you want to travel, I would advise booking tickets now. The best discounts go quickly, and if you can find very low prices, you won't be out of pocket too much if your plans change. (You can't change discounted tickets.) If you have doubts about the trip or its timing; I would wait until your plans are firmer. You can often get some discounted, but not the best, even a few days before travel, There are no discounts at all the day before travel, but two days before travel, you might find decent discounts.

I've just noticed something new on the Trenitalia site: a variation in the price of Super Economy tickets depending on the train and the date. Looking at Rome to Naples, I see the €16.90 price on some trains in June, but a €19.90 price on others. If I look in March, as close as the 15th of March, two days from now, I still see some SuperEconomy tickets left, but they're €29.90. They must be refining their pricing model to make it more similar to airline pricing.

I also see Economy tickets available for tomorrow, although their own explanation of the terms and conditions (in a popup box next to the price) says that they're not available the day before travel. The Economy tickets are €39.90. They now say, which is also new, that you can exchange an Economy ticket for a full price ticket, paying the difference, one time before departure.

Yes it's called running full trains at the most income possible - why sell the cheapest fares on trains that are likely to sell out at higher prices. like bvlenci shows be flexible if possible on dates and times and then if see something you like HOP on it.

Palenq, the ticket machines accept American credit cards.

thanks bvlenci!