Joomla 3.x comes packed with jQuery by default, therefore you do not need to import your own copy of it. The following code will import it in noConflict mode if it doesn't already detect it:
JHtml::_('jquery.framework');
I think the issue you're having mentioned in your question is that you have 2 instances of jQuery being loaded and therefore it causes conflicts.
Joomla 3.x also has jQueryUI packed with it. So your final code will look like this:
JHtml::_('jquery.framework');
JHtml::_('jquery.ui');
JHtml::_('script', 'media/com_szakdolgozat/js/typeahead.bundle.js');