I want to figure out if we can create automated email sequences in CiviCRM. So, for example, if someone signs up for our newsletter, they get an email immediately that has some of our "best of" information, for example, and then 1 week later, they get another email that has some information about the impacts we're having in the community, but it's all automated through CiviCRM. We have two automated buttons that appear whenever anyone is added to our database or signs up to volunteer or to attend an event. The buttons ask if the individual would like to receive our newsletter and the second button asks if they would like to be sent volunteer opportunities. We'd like to be able to craft an email that will be generated and sent to everyone who selects 'yes' on these two options.
As @ericG suggested the CiviRules extension can do this. You can find the CiviRules extension here: https://civicrm.org/extensions/civirules and documentation on the extension is here: https://docs.civicrm.org/civirules/en/latest/.
If you want to send an email you will also need the Email API extension: https://civicrm.org/extensions/e-mail-api
In addition to the above suggestion, you could use Scheduled Reminders (https://docs.civicrm.org/user/en/latest/email/scheduled-reminders/)
Scheduled reminders have access to date fields. If you added two date fields to your contact - Joined Newsletter and Wants Volunteer Opportunities - your scheduled reminders could send emails based on time periods based on those dates. Your front end form would need to make the choice "Yes" update a date field - not sure how hard that would be. The benefit of using scheduled reminders might also be integration with the email templates CiviCRM stores with a WYSIWYG editor.
In addition to using date fields from the contact entity, you could consider using custom activity types fueled by a Webform CiviCRM (joined newsletter, wants to volunteer) to kick off the scheduled reminders, which might be useful for reports and tracking joins/removals.
Lastly, you could also check out the pathway of integrating with an email marketing tool like Mailchimp which also has scheduled email/auto-responder features. There is a nifty Civi extension to handle that (https://github.com/veda-consulting/uk.co.vedaconsulting.mailchimp/) but it would be considered an intermediate/advanced implementation to pull off without problems.
Hope this has helped!
You might also be interested in checking out this extension: https://github.com/artfulrobot/chasse/tree/master
Which might help with keeping track of where people are on your onboarding process.