I currently use a category blog to display the next five events on our home page (www.stjudes-mapperley.org.uk). I want them displayed in date order, earliest to latest but, with currently some twenty articles to reorder each time I add a new article, I wondered if I am missing a trick in displaying them? Is there a simple extension I can use to create and display the events so they are displayed in date order? I had wondered about Google Calendar but wanted to see what other people thought?

When you look at the settings for the category blog menu item, you have a pane "Blog Layout Options" on the right side. The "Article Order" there should allow you to set what you want to achieve ("Oldest first"). You also can choose which date it should take for the ordering.
You can set the same things globally in the Article Manager Options in the register "Blog / Featured Layouts".

YEs, but the Article ORder is base don create date or publish date - we could create, say ten articles, on a single day (but not necessarily in the required date order). At the moment, I am using ORdering as the Article Order and doing it manually. I could, I suppose, set up each item in order, starting with the earliest and then use create date but what happens when I add another article into the sequence?

If you add a new article, the whole category gets reordered by J1.6 and your newest article will have the lowest number (0). All others are incremented.
So this is good if you want the new article to be on top. But as far as I see we can't order it by "ordering" descending, only ascending.

If I wanted to have the articles in an order where the oldest is first and the new ones get added at the end, I would use the "created date" and set this date accordingly.

For SermonSpeaker 4.1 I will include a similar behaviour, but I include the the possibility to order them after "Ordering Descending"

why don't you get jEvents component, create series of events on a calendar, and then use modules anywhere component to drop in upcoming events module on your home page, something kind of like what i'm doing on our home page. www.missionarychurchofgrace.org/j/en.