I'm not completely sure I know what you are trying to accomplish. However, it sounds like the things you are talking about are not mutually exclusive.
The serial number is a way to have the database uniquely identify each record independently of every other record in the database, regardless of the data contained in that record.
When you have the serial number, auto-number setup for your table, when you import the data, make sure you have nothing assigned to that field for import and it will take care of assigning the serial numbers to each record as they are created.
Then you related the tables to each other however you need to to make your data relate. The great thing about Filemaker is it makes this easy to follow and understand (Most of the time) but I would caution you, to make sure you document what and why you are doing things with notes in the table layout graph. It might make sense you to today, but in 6 months when you need to look at why something isn't working the way you think, you will have a "What was I thinking Moment" and it will help you to have that thought captured somewhere so you can refer back to it later.
but how do I connect each of the pieces with the finished product?
We've already discussed this in a different thread where I suggested setting up a BOM table for listing all the parts used to construct a given product. It would be easier to answer this question if you used Post a New Answer to post follow up questions into that thread.
Yes, you can select and link in one part at a time using a portal to the BOM table. But no, you do not have to do this if you have files that list all the parts for a given product from which to import this data. There are ways to automate that process. The exact steps will depend on what data is in that excel file and how the data is arranged columns and rows, but a general description of the process is that you'd likely import these records for a single product into a table and then use a script to loop through that data and build the needed records in both the BOM and the Parts table. (And your parts table can also be your products table.)