I'm quite unsure on how far you intend to walk and in which direction ... and I'm always mystified by "caminos" without a definite pilgrim goal in sight -- which might be whichever Sanctuary of course, even one very local, but I've personally always found walking for walking' sake to be somewhat ungratifying.
But then of course I know exactly nothing of your own motivations nor purposes, so autant pour moi.
But I must say that the oldest and most traditional pilgrim Way in that area of the world is roughly the pathway down to Rome pretty much directly North-South from the broad Pass through the Tirol in Austria then onward South.
Though the better variant of it seems to be from Vaduz in Liechtenstein down to Milan, then Piacenza Parma, Modena Bologna, Florence Siena, then that delightful last stretch through old Etruscan Tuscany and the deeply civilised Lazio to Rome.
You say that your purpose is to enjoy your time there ?
In that respect at least, as long as you keep a watch on your possessions like a hawk against thieves, and can manage to suffer some short periods of inconvenience from the presence of Industry in the Pô valley, the Ways in Italy are basically unbeatable in terms of simple pleasures.
Especially once you reach the astoundingly, breath-takingly beautiful Tuscany.