If you are actually on a modest budget and don't have any compelling reason to go to Tokyo don't go there this time.

It's dirt cheap to get to Kyoto and Osaka from Nagoya and from Kyoto/Osaka it's cheap and easy to get to a whole lot of other places.

So, assuming you actually are on a budget - DON'T get the JR Pass

Nagoya Airport to Nagoya City centre is from approx 850yen by train or limousine bus

Nagoya City Centre to Osaka costs from 1,800yen by bus
http://willerexpress.com/bus/pc/rose...d=3&rosenId=20

Osaka to Kyoto takes about 30 minutes by train and costs under 500yen
Osaka to Nara costs around the same

There's accommodation in Osaka from 2,000yen a night (private room with shared bathroom facilities)
There's accommodation in Kyoto from 2,700yen in some traditional town houses.

If you do want to go to Tokyo from Nagoya you can go by bus from 2,600yen (much longer journey)
http://willerexpress.com/bus/pc/rose...d=3&rosenId=12

Accommodation in Tokyo costs from around 3,000yen per night for a private room with shared facilities

One suggestion to keep costs down whilst maintaining some standards would be to slum it from Nagoya to Tokyo on a cheap bus for 2,800yen (journey time 5 hours)
http://willerexpress.com/bus/pc/rose...d=3&rosenId=12

Go upmarket by getting a luxury overnight bus from Tokyo to Osaka/Kyoto http://willerexpress.com/bus/pc/rose...Id=3&rosenId=1
(coccoon or Business class comfort chairs only)
This would cost around 10,000 yen and save you about 3,000yen on accommodation costs

You can get back to Nagoya from Osaka for under 2,000yen (as mentioned above)

Cost of round trip:
NGO-Nagoya (850y) x 2 = 1,700yen
Nagoya to Tokyo by bus = 2,800yen
Tokyo to Osaka by bus = 10,000yen
Osaka to Nagoya by bus = 1,800yen

Total: 16,300yen (which includes one nights accommodation... of sorts)

Take Tokyo out of the equation and the basic travel costs are only about 7,000yen (I've factored in the costs of getting from Osaka to Kyoto and Nara and back again)

If you want to ride the shinkansen bullet train there are budget options for this too. Best one I know of is the bargain rate JR East Pass
http://www.jreast.co.jp/e/eastpass_sp/index.html
Which effectively lets you ride for three days out of 10 at just 3,333yen a day. You can go almost anywhere North of Tokyo but it's only valid from 1st September to the start of December.

If you decide you want to do a lot of local travel around the Osaka/Kyoto area there are different passes and options to consider
Many are listed here
http://wikitravel.org/en/Kansai
Bear in mind that it only costs around 1,000yen for the return trip to/from Osaka and Kyoto. Make sure you aren't paying 2,000yen for a day's travel with one of these passes when you won't be needing it. I've personally never bought one.

Another option to consider as a budget traveller are the wonderful Okayama International Villas http://www.harenet.ne.jp/villa/
These can be reached inexpensively from Osaka and you can stop off at Kurashiki on the way there (Himeji castle is currently under scaffodling undergoing rennovation)

You can check out and compare travel costs from one place to another for yourself using this great tool:
http://www.hyperdia.com/en/
However, it won't show you the cheap bus options I've described and linked to.