As you’d imagine for a big city, for a visitor the best way to get around Tokyo is by public transport. No doubt you can hire a car but I’ve no idea what the experience is like, how easy it is to park etc. I do know it can get congested at times and of course the Japanese drive on the left like the UK.
Taxis are good but not all take credit cards, which given they’re expensive (minimum fare is around $8) means you should have plenty of cash with you. When congested the driver should ask if you want to take the (toll) freeway which could add several more dollars to the fare.
For the airports the best bet is a limo bus (which may even go door to door depending on your hotel) ranging from $15 for Haneda to Shinjuku up to $35 from Narita. There’s also the Narita express train at around $40 which stops at the major stations like Tokyo and Shinjuku though depending on traffic it may be no quicker than the bus. Haneda is served by the monorail but you’ll probably have to navigate the metro to get to where you are going which could be a pain with luggage.
Although Haneda is closer to the city, it has few international routes, though in October a new international terminal opens and multiple international cities will fly direct.
The metro itself covers the city well and can be a little intimidating at first given all the lines and stations. Avoid the rush hour as you will be unceremoniously crammed into the cars. Pricing is a little confusing as it’s run by multiple companies – on one line 160 yen might take you half a dozen stations while on the other, it will cost 170 yen to go a single station. Luckily they do cooperate when changing lines and they wont charge you the full rate on each. Get a Suica smart card to avoid buying tickets for each journey.
Going further afield the Shinkansen – the bullet trains – are perhaps the best bet. Tickets can be bought from machines in the station just like an ordinary train so more convenient than air travel, while covering most of the country. Cities hundreds of kilometers away can be reached in less than 2 hours from Tokyo station.
Finally when you get to where you want to be, Tokyo is very pedestrian friendly. There are frequent road crossings and drivers actually stop for you unlike most of Asia.