I want to know if it's possible to send an email through a form in the website and then we can check that email in the normal email service (like gmail, etc) and the wordpress' dashboard.
Thanks
Yes. You can use Contact Form 7 plugin to create a form that will send messages to your email (like gmail, etc) and use another plugin from the same author, Flamingo, to have those messages saved in WordPress database and available in the dashboard.
If you prefer to avoid plugins, here's what you need:
message
to store submissionsinit
hookmessage
post type with wp_insert_post()
wp_mail()
As you see, plenty of work/code. If you're a developer, you should go this way to tailor it exactly for your needs. If not, I would strongly recommend you to use mentioned plugins (or any good alternatives) to avoid having some troubles with insecure form.