after a vacation in NY I am looking to ride the Amtrak to Boston in May next year. Any hints/tips.

Reserve ticket on amtrak.com, or go to Penn Station and buy ticket. Wait to hear what track your train to Boston is on. Board train; get off train at Boston.

It has been my experience that Amtrak tickets, like airline tickets, go up in price the longer you wait to buy. I always reserve / buy on-line--tickets may be picked up at a station service window or self-serve kiosk.

Sorry, I should have said, we're from the UK. No experience of any Amtrak rides. Is it safe enough? Is there priority on which side to sit? Are there different levels of class seats? we are being offered business class
obly. What station should we arrive in Boston? Staying in Beacon Hill area.

There is a 10% discount if you have AAA and get/reserve the ticket 3 days in advance of travel

Welcome to the USA. Between NYC and Boston Amtrak runs two types of trains each with two classes of service. All trains are reserved so reservations should be made before you get to the station either on line or via phone. You can pick your ticket up at the station using a self serve kiosk or thru the ticket agent. The Acela is a European like train with business class and coach class. It is higher fare (premium) and offers a slightly faster trip with limited stops between NYC and Boston The Regional service is conventional equipment with business class seats and coach class seats. The slightly higher fare business class seats are in a separate car and are usually in 2 and 1 seating configuration. I usually go for the business class seat as I prefer to work on my laptop when travelling as this car offers plug ins at the seat and is less crowded than the coaches. The ride is mostly thru industrial towns and suburban/rural areas with occasional views on the right side of Long Island Sound.
The trains are often crowded on Fridays and Sundays. They are safe as any public conveyance could be.

If you are staying in Beacon Hill area than getting off at Back Bay Station would be only be a short taxi ride to your destination. You did not say when you were travelling?
If business class is only ticket being offered it would mean that the the train is probbaly sold out in coach and I would definitly take them.
There is no assigned seating. On busy days sometimes a ticket doesn't even get you a seat but standing room or seating in the cafe/lounge car.

Some clarification.

Regional Service - 4hr to 4hr20min
Reserved coach (from ~$59)
Business Class (add ~$30)

Acela Express - about 3hr35min
Business Class (from ~$84)
First Class (add ~$66)

Several of the large universities in the Boston area have their commencements (graduations) in May, so it's a very busy season for hotels and visitor-oriented transit. Whatever you decide, book early.

Riding towards Boston, you'll find more interesting views on the southern side of the coach. If you are on board the train, that's on the right side as you face the direction of travel.

Boarding Amtrak at Penn station can be quite boisterous; I'd choose the regional train (vs Acela) but I'd pay a little extra for business class.

If you pay (i.e., heavy tip) a "red cap" porter to get you and luggage to the train, he'll be able to take you down in a special elevator before the madding throngs arrive on the platform. Well worth it if you can find someone to help you.

Be aware than Penn Station has 3 sections - Long Island RR, New Jersey Transit and Amtrak - all with different waiting and ticketing areas - so be sure you're in the right one.

With so many trains in and out of the station none stays long - so be ready to move quickly as soon as you train is announced. You will have about 10 minutes to get to the right entrance, down the stairs and onto the train. It's likely many seats will already be taken - so to sit together move quickly.

If you have more luggage than you can easily carry running downstairs have a porter take it down in advance using one of the elevators.

The seats in coach are perfectly comfortable (2/2 and nicely upholstered) and due to track restraints the Acela is rarely much faster than the regular Metroliner- so IMHO not worth the extra money.