Tokyo to Jigokudani ("monkey onsen") by car

I don't recommend driving from Tokyo to Kanbayashi Onsen in Nagano where the monkey park is. Traffic in the capital is fiendish, finding your way difficult and congestion is frequent and unpredictable. Much better to take Nagano Shinkansen to Nagano and rent a car there, or take public transport from there. See official website.

Some general points Americans need to know:

Japanese law states that you drive on the left, and cars have steering wheels on the right. May take some getting used to.

Many road signs are only in kanji.

You will need an international driver's permit, not just your American license.

If you have an accident, you will likely have to pay for some part of it. There is a general rule of thumb here 80-20, where even if it is not your fault, the accident could cost you 20%.

Tollways are horrendously expensive.

Aside from the tollways, if you choose other routes, you may find yourself very frustrated with the slowness. Many other roads have speed limits of only 50 kph (30 mph) or 60 kph (35 mph).

Public transportation here is very good.

I agree with #1&2. Driving in Japan for the first timer can be quite the challenge. I know because we drove there last month.
Their public transportation is top notch so you're usually better off taking this route rather than getting stuck in traffic, paying expensive highway toll fees and ending up on the opposite side of the road.

If you're up for the challenge, I would recommend the following:

-MUST get a car navi. They're programmed in Japanese and as far as I know, there isn't an English option. However, you can make out the directions from following the arrows on the screen. If you understand Japanese, it is a plus as the car Navi will always inform you ahead of time as to what direction to turn and what the exiting toll fees cost.

- There is a slow lane and a fast lane on the highway as they do in North America (however it is the opposite to here...the passing lane is the inner lane). Unlike here, people are very couteous about the rules so if a car is coming up behind you on the passing lane, you should move over. Here you'll find that people hog up the passing lane thinking that they actually drive fast and don't give a hoot if cars are lined up behind them but this is not the case in Japan.

-Always have a list of phone numbers of your destination handy. You'll either have to input a phone number or address into the car navi for it to route your map.

-Perhaps rent the car for a day or 1/2 day to see if you're comfortable driving as you wouldn't want to commit to a rental contract and have to opt out short.

-Be careful when you make a right since that's when you sometimes end up on the wrong side of the road.

-Not 100% sure as no one mentioned the road rules to us but I don't think you can make a left turn on a red light.

It takes some getting used to even if you speak and read Japanese (like me) but it was a pretty adventurous experience. I can't discourage it 100% but be up for the challenge.

Take the train to the nearest major station and rent a car there. JR rent-a-car, among others, is at all major stations. In fact, if you rent a car from JR, I believe you get a modest discount on the rail ticket!

Once you are out of the big cities, driving is fine. In the big cities, forget it!

Thank you for all your messages.
I was afraid that driving in Japan may be too much of an ordeal and after reading your responses we've decided on taking a train We both love the flexibility of driving a car so we may still decide to rent in Nagano. Anyway, you've been very helpful!!!
Thanks again!