How large is the lawn? It sounds like a nightmare to do what you've stated. If the lawn has any size, I'd be thinking about using a Bobcat to pull the fabric up. But actually, I'd be thinking about leaving the fabric in place under the turf (if the turf can be made decent) and just removing it from any landscape beds I make. Manually. You'd need a decent, sharp, medium weight round point shovel for digging. An overall picture often helps because many people leave out sometimes helpful facts that inadvertently show up in the picture.

Thanks for the reply! This is our yard, it isnt suuuuper big, but large enough that it will be a hassle lol, I don't mind just gradually digging away at it but if there is a way to do it easier/faster then I'd like that!

I would not want to dig up the whole lawn. Where did you want to add beds?

Good points! I will definitely try digging in a few more spots because you are right, the fabric might not be under the whole lawn! I wanted to convert maybe the third of the lawn closest to the sidewalk into an area for planting shrubs and flowers and things, so was already planning to do quite a bit of digging, it would be nice to use a sod cutter if i can , i will have to look into rental options, thanks!

Removing what's under the whole lawn is a can of worms I definitely wouldn't open, as there's no real need. You can improve the grass much without doing that. It would be the worst job for little return.

"I wanted to convert maybe the third of the lawn closest to the sidewalk into an area for planting ..." If you're going to pen in the walk with plantings I think you're already going in the wrong direction. IMO, a better place to expand the bed is toward the right, possibly even connecting it with what is probably the neighbor's hedge.