Yes, Milan looks like the most obvious transit destination for you.
In order to get some idea on schedules you can check at
http://reiseauskunft.bahn.de/bin/query.exe/en?newrequest=yes&protocol=http:& But generally saying - it will take at least a full day in order to get from Innsbruck to Nice.
In order to make less changes and save on budget you should look at two direct EC trains departing to Nice from Milan (
http://www.trenitalia.com/cms/v/index.jsp?vgnextoid=9bd599e8e609a110VgnVCM1000003f16f90aRCRD), for both you can get 15 euro tickets (if purchase in advance).
Getting from Innsbruck to Milan is slightly more complicated, but still you've got couple options:
1) start you journey early in the morning and with changes in Fortezza, Bolzano and Verona arrive to Milan at 13:25 pm. Probably the only price options for you here will be either breaking you ticket via border Innsbruck-Fortezza (app. 10 euro) + getting Fortezza-Milano ticket for your connection online from TrenItalia (32.7 euro) or rather expensive Global price tickets for the whole route Innsbruck-Milan (about exact fare ask ticket counter in Innsbruck - will be somewhere around 55 euro).
2) You can catch an overnight train to Milano from Villach (00:03-8:25). For that train you can get discounted tickets (purchased online from OEBB website -
https://ticket.oebb.at/bin/frame_ticket.pl?ticket=jticket.pl&LANG=EN). In order to get cheaply from Innsbruck to Villach it is better to use Einfach-Raus-Ticket (group ticket - see our Austrian page). Your connection is 17:01-22:05 with a change at Lienz in Osttirol.
p.s. and yes, Eurail passes is fully useless here, since you can make the whole journey for less than 60 euro per person
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