I've been looking at a bunch of SMTP relay services to recommend to users of my software. While e-mail hosting wasn't my primary focus, I came across https://purelymail.com/. It's a bit of a one-man operation (with plans in place if he's run over by a bus), but he's built quite a good system. Two pricing options, but for a small business $US10 per year should cover it.

Microsoft 365 Exchange Online or Google Workplace Business Starter, good value, a huge amount of IT Professionals will be able to help you with it if you get stuck and they're pretty much the industry standard. 

Either option is less than $10 a month. Or you can jump up to Office 365 Business Standard / Google Business Standard if you want to look at file storage etc.

I second the 365 suggestion.  The Business Basic plan at $10/month gives you a very large mailbox, and the email system has a good reputation meaning your emails are less likely to end up in someone's Junk email.

You also get 1Tb of storage for files using OneDrive, and if you turn on the 'known folder backup', you Documents, Pictures, and Desktop items get synchronised up to the 365 cloud storage so if your computer is lost/stolen/fails you can get the data back easily.  (It's not a true backup, but it's substantially better than nothing.)

If you want only the email account and no file storage, then the Exchange Online Plan 1 at $6.50/month might work for you.

https://www.microsoft.com/en-nz/microsoft-365/business/microsoft-365-business-basic

Compare Microsoft Exchange Online Plans Microsoft 365

Back up your folders with OneDrive - Microsoft Support

Google probably has an equivalent, though I'm not familiar with their platform.

I use Fastmail.

You can do this for $6 per mailbox per month with Exchange Online P1

I use Zoho Mail and have been pleased with the solution

We've tried both O365 and Google Workspace hosted domain services and thought that O365 was better. 

Both are about the same price. 

sfrasernz: Thanks for the suggestions.

Could I also ask for recommendations on domain name registration?

Metaname is highly praised on Geekzone. 

turtleattacks:

 

Metaname is highly praised on Geekzone. 

+1 for Metaname. Not the cheapest option, but it has a simple, easy to use interface, that doesn't try to up-sell you crap you don't need.

sfrasernz: I’m starting a very small business ... cost effective hosted email service?

We pay $1.70 per month to Apple For iCloud+

Recommend crazydomains.co.nz+google workspace for email

sfrasernz: Thanks for the suggestions.

Could I also ask for recommendations on domain name registration?

1st domains for your domain registration with support people just a phone call away between 8am - 6pm mon-friday #SlightlyBiassed 🤫

Yoban:

I use Zoho Mail and have been pleased with the solution