There is a train station right in the Frankfurt airport. There are several departures every hour and the travel time is generally less than 90 minutes to Dusseldorf and several hours to Munich depending on the number of stops. Most of the shorter duration connections on LH are via the train anyway so you may as well get used to their timetables. The Munich airport is good but way out of town so the train is good if you have to connect from FRA. Munich has quite a few flights to the US, Dusseldorf a few, even Stuttgart has a couple. I'm pretty sure Venice will require a connection back to the US.http://www.deutschebahn.com/en/start.html
(the dates are European, day/mo/year. There are a couple of dozen Frankfurt stations, Frankfurt(M) Flughafen Fernbf is the airport. Hbf is the main station abbreviation used commonly everywhere. (Haupt bahn hof) bahn being rail and hof being house.