How to reach Limburg by car
If you want to come by car, you can take the highway A3 Köln/Cologne - Frankfurt. Limburg even has two exits on this highway - Limburg Nord (north) and Limburg Süd (south). If you drive from one exit to the other, you pass the bridge over the river Lahn from where you have a nice view on the cathedral. If you want to go to the center, the north exit may be a bit closer, but it doesn't really matter which one you take.
If you come from north east (using highway A45, for instance), you better take the streets B54 (from Siegen) or B49 (from Giessen). It would be take you out of your way if you tried to drive only on highways - besides these streets offer you a nice opportunity to drive through the Westerwald mountains!
Parking in Limburg is not difficult as there is a parking guidance system which shows you where you find a parking and how much parking spaces are available.
How to reach Limburg by train
You can easily reach Limburg by train. There are connections from Frankfurt, Köln (Cologne), Koblenz and Gießen. The railway from Koblenz to Gießen via Limburg is along the river Lahn, really nice!
From Frankfurt, you can take the ICE (intercity express). It's about 20 minutes then (from Frankfurt airport). Unfortunately they run only every 2 hours. And please note that the ICE station is outside of Limburg. Do not confuse it with the station in the centre of Limburg! There are busses from the ICE station to the center, though.
You can also take a "usual" train from Frankfurt. From Frankfurt main station, it's about 1 hour drive, and you will arrive in Limburg at the station in the centre. Direct trains usually run once a hour (sometimes more often). It's also possible to take the S-Bahn number 2 to Niedernhausen and frome there take a train to Limburg.
From Köln/Cologne, the fastest way by train is to take the ICE. It takes about 40 minutes and you'll arrive at the ICE station in Limburg. Trains run about each hour. Another possibility is to go via Koblenz, but although you'll see nice landscapes, I'd rather prefer the ICE - it's much faster!