Hi,
The 7 JR companies used to be the government owned rail company, so they went to nearly every major city and town in the country. If it has a popular tourist area, probably you can get there on JR. But if you want to check if you can get there on a JR line or not, it is very easy on Hyperdia - you simply click on "More Options" and uncheck "Private Railway". If you do a search and the rail line appears in purple, then you also know it is not JR. Note however that there are a few cases where you can get from one place to another on JR (such as 2 places divided by mountains) but that is not always the fastest route. It may in fact be better in a few cases to pay for a bus to get you there faster to save several hours of time - this is sometimes the case in Nagano, for example.
Best of luck.