First things first, hands down, the best way to move around Japan, be it inside cities or between them, is by rail. Japanese love their trains and it exalts everywhere. Then, when one speaks about transportation in Japan, specifically in Tokyo, queue the huge convulsed map of the metro, and then the anxiety of trying to figure that out. We are not going there.
What you really need (should) to know is that if you want to go from A to B, there is probably a rail that covers it, or at least a bus line. Even in remote areas or islands, transportation is not a big problem: it usually exists, but might run at reduced timetables.
Below is a map of Japan with the main lines. This is it, no need to panic over a convulsed map. From any city along the lines (only major ones are marked), there are rails branching to nearby cities, sometimes even parallel slower speed trains. You can also travel the country by bus (most notably, the Willer Expresses night busses, that even offer a Pass for discounted unlimited travel).
Even in Tokyo, with its intricate web of subways, over-ground metro and rails and busses, things get a lot calmer if you just learn the 3 basic axis lines: The Chuo-Sobu line for West-East, the Keihin-Tohoku for North-South, and the Yamanote Loop Line for downtown, as shown in the map bellow.
Note that without all the overlay of streets and other subways, bridges and buildings, it doesn't even look convulsed. From every station in the image, other lines branch out. Note how the Yamanote Line almost delimits the downtown hub bordered by Ikebukuro on North, Shinjuku on West, Shinagawa on South and Tokyo on East.
Osaka also offers a loop line, and Kyoto have subways branching out on every way from the central station. Also don't ditch LCC Air travel. Sometimes it is a lot cheaper (and faster) to take the air route between more distant routes (not to mention the ones that are only feasible by air, like Okinawa).
JR Passes Calculator
(includes bus routes)
The transport system is efficient, pontual and clean, and usually extremely easy to use even if you know not a single word (spoken or written) of Japanese. For cost savings, you have a wide range of mostly Japan Rail passes (the main JR-Pass that covers the whole country being the most known, but there is also a range of regional passes from JR-Kyushu, JR-West and JR-East, with the benefit that some of these can be purchased in Japan). There are also some other deals from private rails that are more present in certain towns. For subways and local lines, there are again some passes, and the IC Cards (Pre-paid cards) that while usually not offering much on the savings, are really convenient in the way you can just pass them on the blockade and go, without having to digure it out prices and buy tickets for each trip (given, obviously, your card has balance).
And if all else fails, there are always cabs or uber around. While there is a fame of cabs in Japan being expensive, it is mostly because sometimes people want to cover vast distances with them. A normal 10 minute ride is not going to put you in a lifetime of debt, but then again, some planning will get you withing 10 minutes walking distance of pretty much anywhere in big towns by using public transportation.
A word of advice: try to figure out your way without trusting apps. Sometimes apps will give you the cheaper (but not more convenient) route, or make weird line transitions. Japan rail system is simple and you shouldn't need to be afraid. The worst that can happen is a station that covers lot of lines being bigger than you imagine, and you will need to change floors or even buildings to switch lines, but signs are aplenty and you will hardly get lost.