The Nankai group offers a transportation package that includes round-trip travel to Koysan by train and cable car, plus unlimited travel on Nankai busses within Koyasan. The package also includes a 10% discount at souvenir shops, 20% discounts on entry to several sites, and a free cup of tea at Kongobuji. (You first must pay to enter Kongobuji - but there's a discount for that, too.) We paid 3,280 yen from Rinkuu Town, which is the station near where I live, but also the first stop after leaving Kansai Airport. Prices vary, depending on the station you originate from, but are roughly the same as the round trip train fare alone. The train fare is good for two days, in case you want to spend the night at a monastery.
The Cablecar that takes you up to Mount Koya is the longest in the country. From the station at the top, you then take a bus up to the holy areas. If you want to see everything or want to travel on the women's trail, get off at the first stop (Nyonindo-mae). The Nyonindo and Otakejizou are located here. You can easily walk to the other sites. The entrance to the women's route is also at this first stop.
From Namba station, you can catch a local train to Koyasan, or a limited-express train, which costs a little bit more. It's very reasonably priced, I remember it was only eight bucks or so. You might have to transfer at Hashimoto station, but it's a small station and an easy transfer. Once you get out of the city the ride becomes very scenic, with many dramatic peaks and valleys and lush foliage (maybe dependent on time of year). The cable car to Koyasan is usually included in the ticket price, and there are signs at the station directing you to the car's platform. The cable car is also a great (but short) ride, a really steep climb and a good view. Once you get to the top, there's one more leg of your journey. You must take a bus into the centre of town. This isn't difficult and there is a help desk where English is spoken. Have fun!
There is a network of buses plying between Koyasan station and the two ends of Koyasan - Daimon on the western end and Okunoin on the eastern end. Stops are announced in both Japanese and English. Bus stops are clearly indicated in Japanese and English. The bus services are prompt and efficient. However, if you are making a day trip to Koyasan, do take note of the last bus back to Koyasan station and the last train out of Koyasan. Otherwise, you will have to spend your night at a shukubo and this does not come cheap!
If you purchase the Koyasan Free Pass, it includes a 2-day unlimited bus pass. However, do note that you will have to surrender the bus pass when you take a return bus trip from Koyasan to Koyasan station so don't return to Koyasan station unless you want to leave Koyasan.
To get to Koyasan, take the Nankai Koya Line from Nankai Namba station in Osaka. For best value, purchase a 2-day Koyasan Free Pass for Y2780 which includes round trip by train to Koyasan, cable car tramway from Gokurakubashi to Koyasan station and unlimited bus rides within Koyasan for 2 days. It also gives you reduced admission to certain attractions like Kondo, Kongbuji Temple and the Konpon Daito.
When you reach Koyasan station, hop onto a bus for your destination.
If you reach Koya-San by train, you will finish the ride in this cable-car, at first on a moderate hill, then on a steep one. Just ask for information from Osaka to go to Koya-San, they will tell you about the non-JR train line to take, and about the cable-car, then the bus to reach down-town.