Rail Pass?

The Japan Rail Pass goes up to 21-days. My current plan is to stay for 30. Is my only alternative to buy the 21-day pass, then pay for each train trip after the 21?

Depends on what your exact itinerary is at the moment?

For example if you plan to spend your first week or so in one city then going to another city, then there might be a whole week (or 5 days or so) where you really don't need to use the JR Pass.

Depends on your itinerary, do you have a tentative itinerary to post and see?

First, it's not easy to let a 21day JR Pass pay off. You must make very long distance travel(longer than Sapporo in Hokkaido to Kagoshima in Kyushu Isrand), moreover JR Pass is not valid for subways and private railway.

If your itinerary is like staying in tokyo for a while and traveling around there, moving to kyoto and stayin there for a while, moving to Hiroshima and stay for a few days and traveling back to tokyo, 21days pass never pays off.

Currently, my plan is to stay somewhere in Tokyo, and sightsee across Japan, maybe spending one or two nights in different cities. I wonder if the rail pass pays off then, if I plan to use Tokyo as my primary residence, and use it to travel to other scenic points in Japan.

To make a JR Rail Pass pay off, need to ride on a Shinkansen train about the distance of a Tokyo to Shin-Osaka and back each week the pass is valid, or ride the Shinkansen often (or everyday for fun!) to a different city. For a 21 day JR Rail Pass, starting in Tokyo, one week in Kansai, then one week in Kyushu/Chugoku(Western Honshu), then the last week in Tohoku/Hokkaido will pay off very well with a return to Tokyo. But if your primary residence will be in Tokyo and each day you head out to a different city, maybe too much time will be spent riding on a train, unless riding on a train is the goal. A nice way to spend 30 days in Japan is to move your primary residence around Japan about a week at a time, then less time is spent riding on trains.

Currently, my plan is to stay somewhere in Tokyo, and sightsee across Japan, maybe spending one or two nights in different cities. I wonder if the rail pass pays off then, if I plan to use Tokyo as my primary residence, and use it to travel to other scenic points in Japan.

You can maximize your pass as well as bugetize money by splitting the trip into regions and exploring each area from a main base in that region. For example if you spend your first 7 days exploring Tokyo and the area around Tokyo, then a Japan Rail Pass isn't needed. After the first 7 days in the Tokyo Metropolitan region, activate your JR pass and head onto a different region and base and explore off from there (eg base yourself in Osaka or Kyoto for those trips, or trips to places in those areas).

Also this way you're not losing a lot of time just riding trains from Tokyo across Japan to sightsee for a day then returning back. For example Tokyo - Osaka and back in one day is over 6 hours of just riding on the train.

But again, it depends on which particular places you plan to visit.

I agree with the above- Japan is probably a bigger country than you think, so you will not see much other than the scenery from the train if you come back to Tokyo every night. If you want to see a lot of the country, spend a week in the Tokyo, a week in the north e.g. Hokkaido, a week in Kansai and a week in Kyushu for example.

It is quite often far cheaper to buy individual tickets than have a JR pass, particularly if you are spending a week in the Tokyo area for example.

it's not easy to let a 21day JR Pass pay off.; maybe..but it certainly can as well: we trained (sic: sorry) ourselves to think in yen, and base ourselves in only a few places, and do day trips: using Hyperdia, we certainly would have paid off 57000Y: we spent 3 days in Tokyo, so did the NEx on Day 1, then Yamanote line (not much in Yen there, admittedly), then went Shinagawa Kanazawa (2 nights)Kanazawa Osaka, 8 nights and numerous day trips: Hiroshima Miyajima, Nagoya, Kyoto 3 times, Nara, then back to Narita from Osaka on Day 14; we worked out that was 70000+: on a 45100Y 14 day pass we did very well, and had we more time to spend longer in places (eg Miyajima area) a 21 day would also have been fine.
It really does depend on the itinerary: you can well and truly make any pass worth its weight in gold if you plan well