If I needed to support older (but not ancient) browsers (and wasn't using jQuery), like you are saying, then I would do something like this.
CSS
.hide {
    display: none;
}
HTML
<select id="mine">
    <option value="0">Test1</option>
    <option value="1">Test2</option>
</select>
<div id="display1">
    <div>
        <p>Some Text in 1</p>
    </div>
    <div class="hide">
        <p>Some Text in 2</p>
    </div>
</div>
Javascript Cross Browser Support Code
(function () {
    var slice = [].slice,
        nativeTrim = ''.trim,
        trim,
        classList;
    function isFunction(arg) {
        return typeof arg === 'function';
    }
    function isString(arg) {
        return typeof arg === 'string';
    }
    function handler(object, evt, func) {
        var ret;
        if (evt) {
            ret = func.call(object, evt);
            if (false === ret) {
                evt.stopPropagation();
                evt.preventDefault();
            }
        } else {
            window.event.target = window.event.srcElement;
            ret = func.call(object, window.event);
            if (false === ret) {
                window.event.returnValue = false;
                window.event.cancelBubble = true;
            }
        }
        return ret;
    }
    function addEventListener(object, type, func) {
        var uid = type + ':' + func,
            euid = 'e:' + uid;
        object[euid] = func;
        if (isFunction(object.addEventListener)) {
            object[uid] = function (evt) {
                handler(object, evt, object[euid]);
            };
            object.addEventListener(type, object[uid], false);
        } else if (object.attachEvent) {
            object[uid] = function () {
                handler(object, null, object[euid]);
            };
            object.attachEvent('on' + type, object[uid]);
        } else {
            throw new Error('Handler could not be added.');
        }
    }
    if (isFunction(nativeTrim)) {
        trim = function (text) {
            return nativeTrim.call(text);
        };
    } else {
        trim = function (text) {
            return text.replace(/^\s+|\s+$/g, '');
        };
    }
    if ('classList' in document.body) {
        classList = {
            contains: function (node, className) {
                return node.classList.contains(className);
            },
            add: function add(node, className) {
                node.classList.add(className);
            },
            remove: function (node, className) {
                node.classList.remove(className);
            },
            toggle: function (node, className) {
                node.classList.toggle(className);
            }
        };
    } else {
        classList = {
            makeRegex: function (className, flags) {
                return new RegExp('(?:^|\\s)' + className + '(?!\\S)', isString(flags) ? flags : '');
            },
            contains: function (node, className) {
                return !!node.className.match(classList.makeRegex(className));
            },
            add: function add(node, className) {
                if (!classList.contains(node, className)) {
                    node.className = trim(node.className);
                    if (node.className) {
                        node.className += ' ';
                    }
                    node.className += className;
                }
            },
            remove: function (node, className) {
                if (classList.contains(node, className)) {
                    node.className = trim(node.className.replace(classList.makeRegex(className, 'g'), ''));
                }
            },
            toggle: function (node, className) {
                if (classList.contains(node, className)) {
                    classList.remove(node, className);
                } else {
                    classList.add(node, className);
                }
            }
        };
    }
    window.$ = {
        addEventListener: addEventListener,
        classList: classList
    };
}());
Javascript Working Code
$.addEventListener(document.getElementById('mine'), 'change', (function () {
    var children1 = document.getElementById('display1').children,
        length1,
        index;
    return function (evt) {
        for (index = 0, length1 = children1.length; index < length1; index += 1) {
            $.classList.toggle(children1[index], 'hide');
        }
    };
}()));
On jsFiddle
Update: Adding options
CSS
<select id="mine">
    <option value="test1">Test1</option>
    <option value="test2">Test2</option>
    <option value="test3">Test3</option>
</select>
<div id="display1">
    <div>
        <p>Some Text in 1</p>
    </div>
    <div class="hide">
        <p>Some Text in 2</p>
    </div>
    <div class="hide">
        <p>Some Text in 3</p>
    </div>
</div>
Javascript
$.addEventListener(document.getElementById('mine'), 'change', (function () {
    var children1 = document.getElementById('display1').children,
        length1,
        index;
    return function (evt) {
        for (index = 0, length1 = children1.length; index < length1; index += 1) {
            if (index === evt.target.selectedIndex) {
                $.classList.remove(children1[index], 'hide');
            } else {
                $.classList.add(children1[index], 'hide');
            }
        }
    };
}()));
On jsFiddle
If you were writing without the need to support older browsers, then it would look like this.
Javascript
document.getElementById('mine').addEventListener('change', (function () {
    var children1 = document.getElementById('display1').children,
        length1,
        index;
    return function (evt) {
        for (index = 0, length1 = children1.length; index < length1; index += 1) {
            if (index === evt.target.selectedIndex) {
                children1[index].classList.remove('hide');
            } else {
                children1[index].classList.add('hide');
            }
        }
    };
}()), false);
On jsFiddle