From my limited experience:

Pilates is better if you’re hypermobile or prone to injury, you’re more stable and supported. The downside is the cost is higher and places are fewer/may be less conveniently located.

Yoga is cheaper and more plentiful but you have to watch more that you don’t overextend or twist funny.

Pilates can be done without equipment but it is good to have an exercise band and a mat for use at home. There may be other simple cheap pieces of equipment like a dumbbell, kettlebell, you could even buy an exercise ring from Kmart. But if you did it at a place (as I did for physio) they have equipment.

Yoga requires much less equipment if you don’t need any aids like blocks or extension bands. Mostly just the mat.

For both you would probably want a soft thick mat with plenty of grip. Kmart is fine for this too. I have both a soft thick mat and a thinner one that’s more for grip, but the thinner one on a hard floor is uncomfortable on my knees and spine.