multiple databases and layouts

I now have multiple databases

I think that you mean that you have many files to your one database. A "database", in FileMaker context, is a group of one or more files each of which can contain one or more tables--all linked in relationships.

I see no reason that your new database should consist of more than a single file with multiple tables defined inside that one file. You can import multiple times from your source files by opening a FileMaker File, Selecting Import Records | File from the File menu and specifying "New table" as the source table and your import will import the data into a new table defined to store the imported data. You can then go to Manage | Database and further modify the design of this table as well as link it to other tables in relationships.

When trying to merge (import) in data from another database in order to simplify.  I run into a problem.  The information I am trying to import, has multiple pieces of information based on one file component.  So my main indentifier is UNITID.  In the database I am trying to import, I could have up to 30 pieces of information based on an individual UNITID, and it will only import one of them.  I am unsure of how to accomplish what I need.

Sorry, but your description of what you have is a bit vague and thus I am not 100% sure that I know what it is that you are trying to accomplish. Best guess, is that UNITID is a field in your Access tables that uniquely identifies a specific entity in your database. In one table, it is probably set up in Access as its primary key and this field then matches to foreign key fields in other related tables in the Access database.

What is not clear is if you have multiple files of data that belong all in one table or multiple files that represent a set of tables--one would be the parent table with this field as primary key and the others related "child" tables with UNITID as the foreign key (Match Field) for which you will need to establish a relationship link in Manage | Database after you have imported the data.

Either import operation is quite easy to do once you know how. To import more records into an existing table, Open your FileMaker file, go to a layout based on that table and select Import Records. (The layout you select, selects the table that will receive the imported data). To import records into a new separate table, use the same Import Records menu option, but it does not matter what layout you select and you pull down the target table drop down and select the "new table" option so that the import creates a new table for your data.

and after I import, I can delete those other databases?

Yes, but I'd do some careful checking to make sure that my import produced the results that I want/need before deleting anything.

nor do I have any experience with databases
I already created layouts within my database, by creating a new layout and selecting "show records from"

Those two quotes definitely go together! But that's OK. A primary function of this forum is to help out "newbies" such as you, but I strongly recommend that you search out some additional training resources on FileMaker and relational database design. That can provide you with broad based support where this forum works best as a way to answer specific questions.

A layout, a table and a table occurrence can all have exactly the same name and when you create a new table in FileMaker, the system automatically creates a layout and a table occurrence of the same name, but these are three different things in your databases with three different functions. A layout, specifies a table occurrence in that "show records from" drop down. This is the name of one of the "boxes" found in Manage | Database | Relationships. This "box", in turn, refers to one of your tables in the file. The table occurrence that you select for a given layout establishes a "context" or a "starting point" from which to control access to records in related tables. So your layout gives you direct access to records in a specific table and possibly enables indirect access to records in related tables. If you edit data in a field on a layout, that data is from a field in either the layout's underlying table or one linked to it in a relationship. Any change that you make to that data will be automatically saved back to the table where this field was defined.

For more on Table Occurrences, see: Tutorial: What are Table Occurrences?