Let’s take an example. What’s the difference between “I regret to tell you that your father died” and “I regret telling you that your father died”? Well, “I regret to tell”, “I regret to say” are used to inform someone of bad news in a polite way.

Before you tell someone bad news, it’s polite to warn the person that he is about to get some bad news by telling him you feel sorry about it. “regret to” must be followed by a verb which means “say”, for example, “tell”, “inform”, “announce”, “report”. Verbs that do not have this meaning cannot be used.

On the other hand, “regret doing something” means you wish you had not done something. So, “I regret telling you that your father died” means you wish you have not told someone his father died.

Maybe, because after hearing the bad news, he tried to kill himself. Or maybe because it was a mistake and his father had not actually died.
Let’s look at another example. “I regret to inform you that the basketball game has been cancelled.” In this sentence, I’m telling someone some bad news.

 The bad news is that there will be no basketball game tonight. Here’s another example. “I regret eating so much.” This sentence means that I ate too much in the past, and now, I wish I had not eaten so much.

So, if you have to tell someone bad news, say “I regret to tell you…” But if you wish you had not done something, say “I regret doing it.” And if you are glad that you did something, say “I don’t regret doing it.”