Edit in JSFiddle

<button id="button-1">Кнопка 1</button>
<button id="button-2">Кнопка 2</button>
<button id="button-3">Кнопка 3</button>
<div id="slider">
<div><img src="https://bipbap.ru/wp-content/uploads/2017/04/leto_derevo_nebo_peyzazh_dom_derevya_domik_priroda_3000x2000.jpg" alt=""></div>
<div><img src="https://s1.1zoom.ru/big0/788/Lavandula_Fields_Blue_525422_1280x778.jpg" alt=""></div>
<div><img src="https://cs8.pikabu.ru/post_img/big/2017/06/06/2/1496710261194222281.jpg" alt=""></div>
</div>

              
	$('#slider').slick({
		slidesToShow: 1,
    	slidesToScroll: 1,
		adaptiveHeight: true,
        dots: false,
    	prevArrow: false,
    	nextArrow: false,
    	fade: true,
        speed: 900
      });
	$('#button-1').on('click', function() {
	  $('#slider').slick('slickGoTo', 0, true);	
	});
  $('#button-2').on('click', function() {
	  $('#slider').slick('slickGoTo', 1, true);	
	});
  $('#button-3').on('click', function() {
	  $('#slider').slick('slickGoTo', 2, true);	
	});