Oita has its own airport. Tokyo is an hour and half away (plane) and Osaka 50 minutes. The Shinkansen doesn't go through Oita but you can take it as far as Kokura (Kitakyushu city) and transfer to the JR line. The Sonic and Red Limited Express trains run from Kokura to Oita every hour.
If you plan to do a lot of travelling in Japan by train then there is a Japan JR Rail Pass available to foreign travellers. See the JR East website for details:
It must be bought abroad and stamped in Japan before you can use it. There are several options including one week (unlimited JR travel except for the super fast Nozomi service), two week, JR East and JR West passes. Check with your local travel agent for the best option or click on the link to see for yourself!
Here is a link to a very cool page that lets you see the routing between cities/stations/ports within Japan. It will tell you the route and includes the cost and time. http://www.jorudan.co.jp/english/norikae/.
There is a domestic airport in Oita prefecture so you can fly into Oita from all over Japan but I personally recommend to take trains because it's cheaper.
If you are coming in from Honsyu Island, Osaka or somewhere, then take the Bullet Train, shinkansen, to Kokura Station then change to either local train or express train. The first city if you take train to come into Oita is Nakatsu city. By local train, from Kokura to Nakatsu is about 1hour. By express train is about 30 min. If you want to get into Oita city straight away then I strongly recommend you to take express train. Kokura to Oita by express train is about 1 hour and a half. The trains are not bad to sit on so you can also enjoy the mountain seenaries.
If you are driving a car, Oita is accessible from anywhere. Just get one map and off you go!
You can even board on a ship and come into Oita. Beppu is a port city and there are many ferries coming in and out between Beppu and Osaka, Kobe and other major cities. You can check it out!
Driving a car is crucial to hang around within Oita. Oita prefecture only has few train lines so it limits your travel experiences. When you see the map, you can see that.