Activating image and video carousels
From a once over:


Magic constants should go on top ( 5000 and 12000
You use this construct .html($('#slide-content-'+id).html()); 4 times, a helper functions might be in order.
You don't need e in the event handling function, so you can just drop that
Image and video carousel use pretty much the same code, that could be a helper function as well.


So something like this potentially:

var imageCarouselSpeed = 5000,
    videoCarouselSpeed = 12000;

startCarousel( '#image-gallery-carousel', '#carousel-caption-text', imageCarouselSpeed );
startCarousel( '#video-gallery-carousel', '#carousel-caption-video-text', videoCarouselSpeed );

function startCarousel( carouselSelector, textSelector, speed ){
  //Start bootstrap carousel
  $( carouselSelector ).carousel({
    interval: speed
  });
  //Set the first caption
  $(textSelector).html($('#slide-content-0').html());
  // When the carousel slides, auto update the caption
  $(carouselSelector).on('slid.bs.carousel', function () {
    var id = $(this).find('.item.active').data('slide-number');
    $(textSelector).html($('#slide-content-'+id).html());
  });
}


I would agree with Larz a jsfiddle or jsbin would have gotten you a code review much sooner.