As another non-techie user of Outlook, in the past 2 helpful "solutions" or work arounds have helped me.
1. Set up a web-based, (not initially Outlook based) e-mail for yourself on gmail. Send "test" e-mails to the recipients you are having trouble with from this new e-mail account. If they get through OK, then the issue is between your service provider and the recipients service provider.Then, get the recipients to send you an e-mail or two, to both your e-mail addresses. . If they work, it might solve your problem?? If not, If you are with, say, BT, then call them.
I am told that with the right " help" from Mr Google, one can run multiple accounts from your Outlook. (I don't)
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2. Go in to your email account via the url, not Outlook. Try sending test type e-mails again. Experiment as with 1.
And don't forget IT 101. Turn everything off. Leave for 10 mins, and turn on again! I was once told that turning everything off sometimes requires unplugging from the wall socket! Being ignorant, I will try anything!!