Can I link multiple forms to a single initial form? For example for an upcoming training, we have 3 days from which applicants must choose one (each

It looks like the rest of your post did not go through or the title did not allow for enough characters to include the full description.  In any case, it is not possible to combine three different forms to have their data aggregated into a single location.  You can, however, have the contents of all three forms in a single form, then let users update their submission as they go:

https://www.jotform.com/help/40-How-to-Let-Users-Update-Their-Form-Submissions-at-a-Later-Date

If there is a different process you are looking for, let us know a bit more detail and we will see what we can do.