Certainly do not consider this an advertisement, but I bought my new domain through the Google Domains beta and set up the email redirection through their site and it was all pretty trivial to do
I imagine you could do the same with other domain providers, but it might be a bit more low level
I think you need to interact with the ‘Treat as an alias’ box when you add another Gmail address to your account.
You also need ‘Always reply from default address’ ticked too, on the Accounts settings page.
Does this link help?
It was actually Google Domains I purchased through - although quite some time ago. This might make the whole thing much easier 
Don’t know if this helps, but with gmail, if you have a . in between your name, for example .@gmail.com, you the same email without the . still works. I’ve done it to register new accounts on services that don’t allow multiple email logins, but that I still need to access via my main email. Dropbox for example. New login works and emails go to the same email.
Also, you can add dots as you please. So abc.def is the same as a.b.c.d.e.f
Oh sweet, didn’t know that…
I’ve known people use it to really take advantage of register you’re email to get something free deals.
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