First time abroad traveller

Hi,
Japan is one of the safest countries in the world and there is no reason to fear for your safety like back at home. You can walk down nearly any street at any hour and feel perfectly safe. Of course you should not throw away common sense, but you should do just fine.
If you go straight from the airport to Osaka, a 7 day pass would save you some money if you can change your plans a bit and return to Tokyo on July 29. If not, then buying regular fare tickets or flying on a low cost air carrier would be the way to go.

Best of luck.

If you choose not to go for the JR Pass, then some regional passes might help save you some money. Seeing Himeji Castle for example is one of Japan's very best sights, and using a JR Kansai Area Pass or a JR Kansai Wide Area Pass would help you save money.
Since you are already seeing Tokyo, you can largely skip Osaka during the day. 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.
Nara is often neglected by some tourists, which is a shame. Missing the Todaiji Great Buddha would be a tragedy. Nara Park has a lot of other great places, such as Kasuga Shrine, Kofukuji Temple, and feeding the many deer in the area. A few other great places missed out even by those who go to Nara though are the Isuien
https://www.youtube.com/watch?v=kk-7IoLELio and Yoshikien Gardens. They are gorgeous and not crowded at all.
In addition, there are other numerous great places you can visit in the area. Okayama is not far and it is famous for its castle, plus Korakuen, one of Japan's Top 3, and also the Handayama Garden. If you'd like to get off the beaten trail for a while, the half day hike up to Konpirasan in Kotohira is stunning. Going to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.
Hiroshima also has a number of good places. Miyajima is the most famous, and aside from Hiroshima itself, Onomichi with its temple tour and Kosanji Temple is spectacular. Just beyond Miyajima also lies Iwakuni with its historical bridge and castle on top of the mountain.

Enjoy your trip!