Re: trying to insert micro sd card
Welcome to Android Central! When you insert the microSD card, you should push it all the way in until you feel it click into place, since it's a spring-loaded lock. If it's mounting then unmounting, its unlikely to be due to the card not being all the way in. Chances are that the card itself may have become corrupt. When you removed it from your old tablet, did you Unmount first (in Settings>Storage)? Removing a card without Unmounting (while the device is on) can potentially cause some file corruption.
Try inserting the card into your computer, or another device. Is it recognized there?