There are several convenient train connections from Frankfurt to Saarbrücken. There is one direct connection every second hour that takes 2:35hr. For more information check Deutsche Bahn's website
Ryanair operates several routes from HHN (f.e. to GB, ITA, ESP). Be aware: there is only a sporadic shuttle bus service to Saarbrücken that takes about 2 hours and the ticket costs 20€.
Luxembourg Airport (LUX):
LUX is about 100 km from Saarbrücken. There is a frequent bus shuttle that commutes between Saarbrücken main train station and the train station in Luxembourg (“Luxembourg Gare”). You can take the bus from the airport to the train station in Luxembourg. There are two good bus connections which costs 2€ one way and which run several times per hour: Bus Line 16 (Eurobus) and Bus Line 29. More Information can be found here.
Once you have arrived at the train station in Luxembourg (“Luxembourg Gare”), you can take the “Luxembourg-Saarbrücken-Express” to Saarbrücken train station: . The cost for the one way ticket is 9€ and the transfer takes about 1hr 15 min.)
SCN airport is about 15 km from the city center. But there are only a few scheduled flights to Berlin, Hamburg, London-City, Milan, Rome, Vienna and Luxembourg etc. Be aware: Unfortunately its main airline Air Berlin must ground all flights by the end of October! Anyway, a public bus (R10) is connecting the airport and the train station every hour which takes 30 minutes and costs 2,60€ one way, but there are also group prices for 10 people for 2,30€ per person.
All information is without guarantee and subject to change without notice. You should double-check all posted information before making any booking.