What you are asking ends up pretty sophisticated for a newbie to tackle. I haven't spelled out every single detial either so take a look at this and be prepared to use Post A Answer to post follow up questions.
Games>-----Fields (>--- means many to one)
Games::FieldID = Fields::FieldID (Fields::FieldID is an auto-entered serial number)
would seem to be the basic relationship to schedule a game for a specific field.
The catch here is that checking for availability like you describe would require an additional, separate relationship that checks a given combination of Field, date, and time for availability. You can create additional relationships in Manage | Database | Relationships by using the duplicate button (two green plus signs to duplicate one of the table occurrence "boxes" found on this tab.
The following method assumes that all field reservations are made for the same length time interval. It does not assume that all reservations start with the same pattern of intervals. (One game can start right on the hour, another at 15 after, a third 30 minutes after, etc.)
I suggest defining two fields, TSstart, TSend of type time stamp in games. These record the date and time a game starts and ends. TSend should be a calculation field defined as:
TSstart + //put length reservation time period in seconds here
Choose TimeStamp as the return type. Define a serial number field: GameID. Then define this relationship:
Games::TSstart > GameReservations::TSstart AND
Games::TSstart < GameReservations::TSend AND
Games::GameID ≠ GameReservations::GameID
GameReservations is a new table occurrence of Games that you create with duplicate button I mentioned earlier.
Now define a calculation field, cReservedList as:
List ( -1 ; List ( GameReservations::FieldID ) )
Select text as the return type. Be sure to select "Games" from the context drop down at the top of the specify calculation dialog.
Now you can set up this relationship:
Games::cReservedList ≠ AvailableFields::FieldID (AvailableFields is a new occurrence of Fields.)
With these relationships, you can put a portal to AvailableFields and when you enter a date and time in TSstart, and the portal will display all available fields. A conditional value list can use the same relationship such that you can select a field from a drop down list of available fields.