With a combination of help from the answers from @Mike and @Jordan, I was able to resolve this issue. The "Lots of People" is coming from the highest level of permission inheritance. I figured this but didn't know how to configure that highest level.
Using the "Check Permission" on a specific site, I could see that users "None" listed as the Permission level given still had "factors that affected their level of access.
In the Central Administration of SharePoint and clicking Application Management > Manage Web Applications > [selecting the name of the website] > User Policy, I could see all "domain users" Permissions were set to Full Read. (This is why users could see subsites that they weren't given access to.)
Closing the User Policy window and going to the Permission Policy menu item, I created a new permission Policy Level that only allowed View Pages and Open permissions under the Site Permissions category.
I then went back to the User Policy settings and set all "domain users" to this new Permission Policy Level.
This along with the setting each subsite members to their appropriate Active Directory groups, allowed users into the top level Site Home and their appropriate section's subsite.
Sites and documents shared, by default, still do say "Lots of People" but those "Lots of People" are people in a specified group, not all domain users as before making these changes.
By going to Site Settings > Site Permissions > Access Request Settings and Allowing access request to an appropriate email address, access requests can now be made to other subsites to which users aren't originally assigned.
Thank you for everyone's help!