There are many ways in which you can improve the security of your site.
The first thing that matters concerning this aspect is the web hosting company you choose. If you have a shared hosting plan, then your plan should already include a firewall, anti-spam email protection, DDoS protection, and so on.
Next, based on what script you are using for your site, there are certain actions you can take in order to make your site as secure as possible.
If you use a CMS (like WordPress, Joomla, Drupal, etc.) make sure you always update the main script, themes, and plugins to the latest version. Be careful of what themes and plugins you use, since these represent in most of the cases a weak point of your site.
All the major content management systems also have various plugins and extensions that you can use for improving the security level of your site. There are just too many aspects to cover all these here.
You should also install an SSL certificate on your site, change the default username (admin) and set a strong password for your accounts that have administrator rights.
Make sure you also make a backup of your site's content and database. If something unexpected happens, then you can always install the backup.