Getting to FRA by train:
Frankfurt Airport has two train stations: One for regional and one for long-distance trains. The walking time between those two stations is about 10 minutes. Some long-distance trains call at the regional train station at night.
The ICE long-distance trains connect hourly Cologne, Mannheim, Basel (some trains also go to Zurich), Stuttgart and Munich with Frankfurt Airport. A train change at Mannheim is needed sometimes. There are also direct connections to Brussels and Amsterdam via ICE. Beware that the trains coming from Paris will not call at Frankfurt Airport, you'll have to change at Mannheim.
The Lufthansa Airail service will call at the long-distance train station.
The airport is also well connected by regional trains, called S-Bahn. They provide a frequent transfer (every 15 Minutes) to Frankfurt City and cost 3.40 3.90 per journey.

Getting to FRA by car:
Situated directly near A3 and A5 motorway, it is easy to find the airport. Parking space is available, but very expensive. Expect to pay 25 Euros per day. The "Business Parking" spaces in parking garage P4 is even more expensive. You can pre-book so-called "Holiday Parking" spaces from Fraport itself, but also from private vendors. A bus transfer (included in the price of the parking) from the parking space to the airport is needed as they're far away from the airport.
You can also park your car at nearby train stations, but these parking spaces are not video-controlled and are notorious for burglars, especially if they see a foreign car. But, as you may like to save some money, you can park your car at Frankfurt-Stadion train station (S8/S9 every 15 minutes) or Neu-Isenburg train station (Bus 651 and OF-67 every 15-30 minutes). You can't park at Kelsterbach train station as they are equipped with parking-meters.
Some airport hotels do offer a "park and fly" service with a package of a one-night stay and free parking for up to 14 days.

Getting to FRA by bus:
Germany's bus network is regulated by government, so there are just a few long-distance-busses calling at Frankfurt Airport. The main bus station for all European bus services is located right outside the Main Train Station, accessible by S-Bahn to Frankfurt Hbf.
But there is a nightly bus from Heidelberg to Hamburg which stops at Frankfurt Airport. You can also hop on the hourly shuttle to Frankfurt-Hahn Airport which also calls at Frankfurt Airport.
There are some Lufthansa-owned bus services to Strasbourg and Heidelberg, these ones need to be booked well in advance.

Getting from the airport to Frankfurt Fair:
At fair times, there is a private shuttle bus departing right outside the Terminal at the bus station directly to Frankfurt Fair. It costs about 10 Euros one-way and will take about 20 minutes to reach the fair.
This bus is only in service at large fairs like the IAA or the book fair.