Different person chiming in. I like to start with a small bite just enough to have a hole for the juice to come out. From there I squeeze it a little and just suck out the juice. Sometimes I leave just enough to not have a completely dry tomato. I feel like the juice adds to the flavor. Nine times out of ten that's enough to not make a mess. For the really juicy ones though there's still plenty of juice in he second half of the tomato. So I bite just a bit, suck out some juice, then finish the bite. Like some sort of weird tomato vampire.
Normally I eat a tomato with a little salt sprinkled on each bite. After that first nibble though I pour a good amount of salt on before I suck out the juice. That way the ratio a tomato taste to salt remains yummy. I've also tried tajin instead of salt. Its different. Not bad mind you but I'd say more of a once in awhile combination just to change it up from the reliable tomato and salt combo.
I've always just left it attached to the main part. My family's always sliced the pineapple into disks then cut those into 4ths. You end up with a small part of core per piece.
If you want to get more out of pineapples did you can use the rinds? It's a fermented drink called tepache. It's the rinds of a pineapple, water, and a type of brown sugar called piloncillo. I've never heard of using the core in it but I dont see why you couldn't add it in. I say add because the rind is where the fermenting bacteria comes from.