1. System to Use
CMS
A CMS, or content management system, is generally the way to go with websites. New items are generally placed on top and older items are lower on the page or even archived. Like the title says, it's all about content. Static content is the same in a CMS as it is in an older static site. Really, it is. If you're never going to update the content, see if you can get someone to convert your old Frontpage site to something that looks good on a tablet, because there's not real point in going to a CMS. That said, it is recommended if you want new visitors to your site. Add an article per day and the internet will notice, and then people will notice, and then you will notice more people showing up at your church or school who found you because of your site, not inspite of it.
Popular CMS scripts include Wordpress, Drupal, Joomla, but I have used many more. Most of these are free and open source. The downside is that these systems can be hacked, especially if left on an older version without system updating. I've had to deal with Russians and Chinese so much that I feel like a Cold War politician. I've learned a lot about making sites more secure over the years, but they continue to seek ways in. If you want a little more security but an even tougher learning curve and less adaptability, then you can use a paid CMS like Microsoft's or one of those former business systems reconfigured for other purposes. My school district was paying over $10,000 a year for a system that looked like a toy to me and was not nearly as powerful as a fully-configured Joomla site or even a Google site. I'd stick with free, make backups, adapt to limitations, and save the money to pay someone to post content. Many of the add-ons to the expensive CMSs I've seen are worthless as they are a decade old. Please don't get stuck with one of these systems just because someone can promise to add all users to the system. Use Google if that's you main goal.
Old school
If it ain't broke, don't fix it. If you email and text and Youtube and Tweet, and you have a site that looks pretty and mostly does its job, who cares if it's twenty years old? However, you will have trouble driving new viewers to the actual site, so you'll want to use those other social media outlets well and maybe embed them in your prehistoric site. It'll actually work.
Free
Do not go with a free subdomain site unless you have a good redirect or aren't worried about people actually finding or remembering your site. Other sites would love for your site to be under theirs so that they get more traffic, but it looks cheap (because it is). If you must go free, I'd recommend Google Sites right now, just because Google offers a hosted CMS for free. Keep in mind that Google does not have huge pool of people developing upgrades or plugins, but some already exist. The biggest bonus is really the simple integration of Google Drive into your site. While these documents are not searchable online, even your least adept user will be able to figure out how to update a document in order to update a site. If Google folds its sitebuilding business, it will also probably offer more of an out than some of the other available free options. Also, unlike most of the free sites, Google does not put ads up for you, which could save some embarrassment for churches and schools. A side note: many ads are based on the places the person using the computer has been, so if a church member complains that your Google ad showed up as a Thai dating site, his own searches led to that ad, not your content.