It’s in Providence but The Steel Yard has classes..

If you haven't yet check out the Steel Yard in Providence for blacksmiting classes

https://www.thesteelyard.org/

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This isn't an apprenticeship but you can take blacksmith classes at Chase Farm in Lincoln.

Bill martin the sculpture professor at RIC could teach you if you take a sculpture class. They have a gas forge and coal forge. They have multiple anvils and a power hammer.

Not near Smithfield at all, but at the South County Museum they have a Blacksmith on staff. They also do lessons, though not cheap. If you do take a lesson and become a museum member they have “open forge” days.

As others have mentioned The Steel Yard has classes, and offers open studio nights Thu, Friday, and Sunday after you've taken a basic class. Pretty sure they're one of the only places regularly offering open studio in Rhode Island, aside from South County Museum which offers once a month.

There's two beginner classes with open spaces coming up this very weekend: a forging for charcuterie class for Saturday and a bottle opener class for Sunday. There are a few other beginner classes coming up, however I'm pretty sure a few of them are miscategorized. The Day of the Tentacle class on the 30th has typically been a beginner class, with the Hammer forged hammers class typically being intermediate and requiring around 20+ hours of previous experience. It may be worth reaching out and asking if the Day of the Tentacle class is actually an intermediate class as I found it to be one of the best intro courses they offer due to the instructor very specifically focusing on hammer technique and how to start/manage your fires (plus the end product is one of the coolest things you can make out of an intro course). Not that the other classes aren't just as phenomenal, just that Mark pays extra attention to good technique which is probably the hardest thing to learn.

All the classes may look fairly expensive at a glance, but keep in mind that the instructors are all incredibly experienced and the shop has almost all the tools you could want/need for a shop. Well worth it if you're interested.

Also, there's fabrication nights on Wednesday and Friday for their annual Iron Pour up until the 21st. I know there's a good deal of welding going on but I believe there's a team also working out of the blacksmith shop which may let you get in and get a feel for it for free. I haven't been able to attend a night so not sure exactly how much hands on experience you'll get, but worth it to meet people and get a look at the facility IMO.

Outside RI, Prospect Hill Forge in Waltham is an option for classes. I've been told the instructors there are some of the best teachers (both in smithing, and in general).

Sorry to say I actually knew a smith in Smithfield, but I just checked and he died! I hate it when that happens! (To people younger than me.) ☹

South County Museum has a couple good blacksmiths there, they do classes but the forge is usually open every weekend and once you take one class you can use it whenever it’s open

My Neighbor is a blacksmith. He teaches all the time. We're right in Pascoag.

There's a blacksmith shop in Lincoln that I took a class at a few years ago. Not sure if that's still happening, but it's legit an oooold smithy. https://greatroadheritagecampus.org/hannaway-blacksmith