Milan is not known for its "cheap hotels" esp. if you happen to arrive during Fashion Week or any of the int'l conferences/conventions that take place there. In addition, for sightseeing I think you'll be disappointed. Rather than stopping at such major cities - if you are really looking to save money, go to some place less popular - Verona, Bologna, etc...

Also, FLorence is only about 90 minutes by train from Rome, so you aren't exactly splitting up your trip by staying there.