I've used a similar method to send workflow emails for a ticketing system we now use at work across different departments. We use these workflow emails to notify both sides of the desk how the ticket is progressing. Most of these are automated but we also have one that can be fired manually to send an update to the ticket creator and anyone they have specified as someone who should be notified of the tickets progress through the system.
I haven't used Site Workflows myself so what I am going to describe is using a List Workflow. In SharePoint Designer (SPD) open the site you wish to customise, find the list you wish to add the workflow to and then on the 'List Settings' ribbon click on 'List Workflow'.
Build the workflow up with anything else you wish it to do. The email action can be found either by clicking 'Action' on the ribbon and 'Send an Email' in the 'Core Action' section of the drop down menu, or start typing the command 'Send an Email'. The action will be shown as 'Email these users' Click on the these users part and a dialogue window will be shown to 'Define E-mail Message'. It is the To box you will probably wish to add your user to.
Lets assume you do go for the "Manager" field being a people picker box that only allows a single person to be entered. At the end of the To: line click on the address book symbol. You will want to pick 'Workflow Lookup for a User...' under the 'Or select from existing Users and Groups:' section. Either double click it, or single click and then click on 'Add >>'. 'Data source' will be the current item as that is what you will want the workflow to fire against. Then 'Field from source' will be the "Manager" field. Then for the 'Return field as' I would pick Email Address. Once you have set these three options, click on OK to return to the 'Select Users' dialogue, then OK again to return to the 'Define E-mail Message dialogue'.
You can use a similar method to then create the email body as you said above you wanted to email the contents of the new list item to the manager.
You can set this workflow to fire on item creation, item change, manually, or a combination of the three depending on what works for yourself and the end users. You can also look to create a Quick Step icon on the ribbon if you wanted it to be manual but easier to start than going to the Workflows section first, then starting it from there.
Just to cover off one final point, if you decide to go the route of using a free text field, instead of selecting the "Manager" field and using E-mail address, you'd use 'As String'. This does however require users to have correctly filled in the free text field with an appropriately formed email address.
