My dad had a grey Buick...back sloped down, probably a late 40 something. I remember going to the garage, maybe to meet him when he came home from work. Not sure how I remember that, they later told me he traded that in before I was three. I guess it was for the yellow mid-50's Merc that I remember liking the style of.
I don't have memories of riding in either of those though...guess because back then, Dad worked and Mom was home with the kids and did all the kid-carting. She had a '53 light green over dark green Olds that she loved. I just recall a million trips in that thing and how huge it was inside...there was room to sit and play on the backseat floor for crying out loud! And I recall so many times where we had 4 or 5 kids, or maybe 6 sometimes, in back, and we'd play "crush the end kid" every time we took a turn. (Good thing we didn't have Hooligan's Studebaker I guess!)
Later we had a '58 Plymouth wagon with the pushbutton gears. Mom used to take half the neighborhood to the beach (we lived in Daytona), stay all day...brought a little grill and basically tailgated at the beach. Great times!
Oh yeah, Dad had the jet-exhaust T-bird too. Same deal, sliding steering wheel...'cept I don't recall any backfiring ;-). His was a '62. Cadfaeltex is right, they changed syles in '64. I think the jet style was '61-'63.