Hi,
Yes your itinerary is doable, and generally I discourage having to go through some tour company - it is easy to navigate and get around, the Japanese are generally quite ready to help when needed, and nothing beats the ultimate freedom of going where you want, when you want, and for how long you want. There is no real need for you to waste your money.
One of the first things you should do upon arriving at Kyoto Station is go to the tourist info office on the 2nd floor. They have lots of free maps and info that can help you out. Of course you can make it easier on yourself as well by studying some maps online of the cities you are visiting before your trip - get familiar with the layout as much as you can ahead of time.
For your Day 1: First, if you are extra ambitious or energetic - this part is optional. Get up really early and go visit the Nishi-Honganji and Higashi-Honganji temples - they are very close to Kyoto Station, and free. They open at 5:30 AM & 5:50 AM, respectively, if you're an early riser. Most tourists just ignore them which is a terrible shame - they are some of the biggest wooden structures in the world and have a fantastic appearance.
After that, you can take the subway from Kyoto Stn to Nijo castle, changing trains at Karasuma/Oike Stn and getting off at Nijo Stn. Nijo Castle is right by the station. The JR Pass does not cover the subway, however. When finished, return to Kyoto Stn. and take a JR train with the Pass down the Nara Line to Inari Station. The Fushimi Inari shrine is right across the street, and you can see a good preview of it here. When done, continue down the same line to the JR Nara Stn., and walk 20 minutes to Nara Park. The Todaiji Great Buddha is a must-see, but also be sure to feed the deer, see Kofukuji and Kasuga Shrine, plus two often overlooked gardens, Isuien and Yoshikien.
Day 2 - You can use your pass to go from Kyoto Stn. to Sagano-Arashino Stn. Actually Nijo Castle is along the way, but seeing Arashiyama can take up a full day to get most of it in. First is the historical Togetsukyo bridge which is over 1000 years old, but today is just another concrete bridge with cars and trucks running over it. So pass on over and go see Tenryuji Temple (especially the garden which is exquisite), from there the bamboo grove is next door, and from there it's easy to find the entrance to the Iwatayama monkey reserve (but takes about a 20 minute stiff hill climb). Afterwards, walk northwards and see the various temples. Don't miss the one at the end, the Otagi Nenbutsuji which is one of the most humorous and whimsical places in Japan you will ever visit.
For Day 3, I recommend Kyoto's eastern side, which is really the most interesting part of the city. You could start very early again if you can, and see Toji, which has Japan's tallest pagoda, and is again very close to Kyoto Stn. It's actually a bit SW of the station. Afterwards, there are many of Japan's best places on the east side - Sanjusangendo, Kiyomizudera, Nanzenji, Heian Shrine, the Philospher's Walk, Ginkakuji, and if you can, way up north is the Shugakuin Imperial Villa which is a gorgeous garden and place of royalty. Prior reservations are required though but it is free.
And if you can squeeze it in somewhere, the Kinkakuji Golden Pavilion is an icon of Kyoto and Japan, and well worth seeing. You can easily get through it in 30-45 minutes, tops.
Yes, you can also go to Osaka by the Pass. After the temples of Kyoto close down around 5PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are wonderful.
If you had another day, going to see Himeji Castle would be well worth it - it's Japanese finest and you can take the bullet train there, adding more value to your pass. North of Tokyo also is Nikko, which makes a great day trip.
Best of luck!