Am I better off taking boxing lessons or a class?

If you genuinely want to take boxing seriously I would suggest both a trainer and classes but obviously, as you mentioned, it will cost alot.
It’s a difficult call to make because it depends upon a few factors, your weight and natural ability being a couple of examples, some people benefit hugely from having a personal trainer while others have enough discipline to commit to training and enough knowledge to understand what their training should focus on. Something else to be contemplated is that while a class led by a good instructor will not cost too much a good trainer is generally very expensive (not forgetting that they also commonly develop more thorough training schemes meaning more hours of training/extra costs).
My suggestion ultimately would be to firstly take some classes to get a basic idea of boxing (and to confirm that you have a genuine interest in the sport) then once that’s done to try a few one-off sessions with various trainers to gather whether there is anything significant they can offer you. If so then you may have to rethink some financial plans or attempt to do without.

Definitely take the classes. Working out is easier with a team. Having friends to support you will help you stay focused. Also, the place I go to offers private lessons with the instructor, so we get the “best of both worlds”. You can inquire about this at the place you are considering. If you are serious about it, don’t take classes from any place, research a few. Consider how many champions they have, etc.

I would join a boxing gym. If you find a good fight gym, either boxing only or MMA you will have the ability to choose from an assortment of classes (jab classes, footwork classes, sparing sessions, etc.). Once you get past the basics and your body mechanics become natural try and find a group of fighters (closer to pro the better) that need sparing partners and sit in and take part of their conditionig while they train.

If this doesn’t work for, if you live in an area with limited choices for fight training, then take the class learn the basics and move on to a trainer once you get the fundimentals down.

Sorry about grammar, I wrote this on my phone.