The "to" (recipient) is pretty simple. It's whatever account you want to read the contact requests at. It can be any email address you own.
The "from" (sender) is more complicated and it absolutely matters what it is.
Let's say the recipient and the sender are both gmail accounts. You have to send the email through Google's SMTP server (authenticated with your Google account credentials) or it will be rejected.
Let's say the sender is Yahoo and the recipient is Gmail. Now you have to deliver the email through smtp.mail.yahoo.com, again authenticated with your Yahoo credentials. When it reaches the Google recipient, Google will verify that the Yahoo sender details are correct or in the bucket it goes.
Let's say you don't want to use a Yahoo or Google domain. You want to use a custom domain, like myawesomeblog.com. Now you need an SMTP server that will relay the email for that domain for you, and receive email as well obviously, and that means an MX record in the DNS zonefile of the domain. You will also need to setup an SPF record, and maybe a DKIM record as well. Lots of steps to ensure Google doesn't toss your message or flag it as spam.
Is it possible to use a sender of someone@wordpess.com? I don't know. You would need to have an account with whichever SMTP server wordpress.com has authorized for use.
At a very minimum to get WordPress to send the mail semi-reliably you must use an SMTP plugin. The default PHPMailer, which uses sendmail, which (likely) uses an unauthenticated SMTP server on the localhost, is not going to cut it in 2015.