You have two options with Sitecore 9 Update 1 and onwards:

1. Use Send Email Campaign Message submit action
2. Create your own Send Email submit action

For Sitecore 9 Initial Release only Second option is applicable as there was no Send Email Campaign Message submit action and also EXM integrated so far in that version of Sitecore

You have two options with Sitecore 9 Update 1 and onwards:

1. Use Send Email Campaign Message submit action

This option has couple of drawbacks:

• These emails can be send out only to known / identified contacts. You need to implement custom submit action described in next point.
• You cannot use data typed in form fields straight away in your email message. You need to create custom submit action which will be processed before sending email and will be updating current identified contact. Example of this kind of submit action is here: https://doc.sitecore.net/sitecore_experience_platform/digital_marketing/sitecore_forms/setting_up_and_configuring/walkthrough_creating_a_custom_submit_action_that_updates_contact_details

2. Create your own Send Email submit action

Rodrigo Peplau already described process of creating this kind of submit action pretty nicely here.

In essence you need to:

• Create your custom submit action based on this article - http://www.nishtechinc.com/Blog/2018/April/Send-E-mail-Action-to-Sitecore-9-Forms

• You can also take it directly from his repository - https://bitbucket.org/nishtechinc/formssendmail/overview

• You can also take a look in Sitecore doc page on how to create custom submit action - https://doc.sitecore.net/sitecore_experience_platform/digital_marketing/sitecore_forms/setting_up_and_configuring/walkthrough_creating_a_custom_submit_action

• You need to create your custom Email Template

• You need to setup Form Designer to substitute fields with keywords like {Subject} or {From}

• Send email using MainUtil.SendMail() at the end

For Sitecore 9 Initial Release only Second option is applicable as there was no Send Email Campaign Message submit action and also EXM integrated so far in that version of Sitecore