Turf shoes or sneakers are fine inside. You can wear cleats outside as well, but I never have. Indoor Turf shoes work fine on all three fields.
Flats or turf shoes inside, the same for outside plus cleats, but I'd recommend not using cleats, since they are slippery and the turf isn't super thick... I guess if the cleats were for short firm ground, they'd probably be fine.
As Spencer said, turf or flat soled shoes inside. The House of Soccer at our facility has a great selection and definitely help with that.
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