Public transport to the Ijen Plateau is available, but not as regular as in cities and not that often. You can reach the plateau from 2 directions, either via Bondowoso or via Banyuwangi. Traveling to the Ijen Plateau is fairly easy if time is your friend. If you are short in time, you might consider an organized tour.

Access via Bondowoso
Paltuding is 64 kilometers east from Bondowoso, but public transport won't bring you that far. You can get as far as Blawan, which is still 17 kilometers before Paltuding.
From the bus terminal in Bondowoso you can get a minibus via Sempol to Blawan/Kaligedang. The bus leaves around  in the morning and it takes about 3 to 4 hours. It takes that long because of the road conditions. At for the price to get to Blawan at the terminal before you depart. It should be around IDR 150,000 per person.
If the time or the mode of transport doesn't suite you, you should be able to find a charter - car or motorbike - to anywhere at the plantation. A car will cost you around IDR 600,000 - IDR 750,000. By motorbike it will be approximately IDR 250,000 - IDR 300,000.
Directions to get to Paltuding and the Ijen crater lake
Anywhere at the coffee plantation you should be able to find someone who can bring you to Paltuding, either by car or by motorbike. At Paltuding you can find also motorbikes to take you back to the Plateau, to Bondowoso or Banyuwangi/Ketapang.
Directions to get back to Bondowoso
The minibus back to Bondowoso leaves around 06.00 in the morning and starts in Blawan/Kaligadang.

Access via Banyuwangi
Kawah Ijen is closer to Banyuwangi than to Bondowoso, but the road is very steep and has deteriorated badly. A 4WD is essential. If you want to spend the night before climbing to Kawah Ijen, it is most likely you spend the night somewhere between Banyuwangi and Licin instead of at the plantation. But spending two nights at the plantation is really worthwhile if you are interested in the Javanese way of life.
Directions to get to Paltuding and the Ijen crater lake
A minibus from Banyuwangi will take you as far as Licin village. In Licin you can spend the night in Ijen Resto & Guesthouse or in the luxurious Ijen Resort and Villas. From Licin you either charter a 4WD or a motorbike. If you rent a car in Banyuwangi to bring you to Paltuding it will cost you around IDR 650,000 - IDR900,000. By motorbike it will be around IDR 250,000 - IDR 350,000.
Directions to get back to Banyuwangi
At Paltuding you will find some motorbikes that can bring you back or to any other destination that you desire.

By motorbike
If you are a traveler alone (and on a budget), a good and interesting alternative might be to go "all the way" by motorbike to the Ijen crater lake. By motorbike makes you feel part of the Indonesian community while almost nobody will notice you are a foreigner. Negotiate with your driver about the costs, what is included and not and where to spend the night.