Interesting places close to Tokyo

Can you recommend me some places to visit near Tokyo? I will be visiting Japan for 5 weeks in august/september and 2 weeks will be spent in Tokyo.
I have no idea about what other places are worth spending the remaining 3 weeks in.
First time going to Japan, so it would be great if the way from Tokyo to the recommended place would be simple.
Will be travelling with a suitcase, so I will need to use some kind of transport which allows having that.

Thank you

With three weeks spare. You can visit Hokkaido,Tohoku, Chubu, Kansai Chugoku, Shikoku, and Kyushu. Even Okinawa.

Does it have to be near Tokyo? And if so, why? Does it have to be places you can make day trips to?

Even if that is the case, there are tons of options. Here is a quick Google search result on "day trips from tokyo".
https://www.google.com/search?q=day+trips+from+tokyo&oq=day+trips+from...

There are dozens of good places as day trips from Tokyo. In fact, August is miserable hot and humid, and going to the mountains would be a great idea. A few places are:
Nikko, Hakone, the Fuji 5 Lakes area, Karuizawa, Kusatsu, Izu, Kamamura/Enoshima, Kawagoe, Takao, Okutama, Yokohama, the Ushiku Great Buddha, Nokogiriyama, Minakami, Showa Kinen Park, Shosenkyo Gorge (Yamanashi), and more. Look into getting a Tokyo Wide Pass or Greater Tokyo Pass to save money.
https://www.japan-guide.com/e/e2357.html
https://greater-tokyo-pass.jp

The question given above is valid though - with that amount of time, why stick around Tokyo? The entire country is wide open to you. For long distances, you can fly dirt cheap on low-cost carriers or ANA's Experience Japan Fare.
http://www.jetstar.com/jp/en/home
http://www.flypeach.com/pc/en
http://ana.ms/20Xtf8t

I think you need to start from here https://www.japan-guide.com/list/e1103.html and here https://www.japan-guide.com/e/e2400.html or any guidebook you trust and get some first ideas for your itinerary.

Once you have some ideas and specific questions feel free to ask again.

Enjoy planning your trip to Japan!

OP here.

Day trips: It won't be a day trip. I will have to reserve accommodations in the place I decide to visit on your recommendation.

About sticking to Toyko area.
A friend told me, that you can not take Shinkansen with a travelling suitcase, so I thought I would have to depend on other kinds of transport (which would rule out far away places?). If that is wrong, please correct me. Not exactly experienced with flying, so I am not sure if it is a good idea to fly inside of Japan...
My Japanese is not exactly stellar. That's why I would love to have some easy travelling, I'd end up lost if it were complicated.

I would love to go into the mountains, how hard is it to get there from Tokyo?

Sorry for not being clear enough, hope this made it better.

�tA friend told me, that you can not take Shinkansen with a travelling suitcase ...

Then your friend has not been on trains before in Japan https://www.japan-guide.com/e/e2274.html

Also consider using luggage deliver services (takuhaibin). They are so convenient and reliable.

If your friend said that you cannot take luggage onto a shinkansen - then your friend has no idea what they are talking about.

Small and medium sized luggage can be placed above the seats. Larger luggage can be placed behind the last seat on the carriage. I find that a large hiking backpack (80 litres of so) fits easily on top of the seats and you have plenty of space to have 1-2 weeks of clothes and some other stuff as well.....

First, you can take a suitcase on the shinkansen. If your suitcase is large, it might not fit overhead and so it might need to be in your very generous leg space then instead.

Second, if you want to travel luggage free there is a wonderful service called takyuubin. https://www.japan-guide.com/e/e2278.html

For me personally, it usually costs about 2000 yen per bag, but they will take your bag from your lodging in say Tokyo and deliver it anywhere from the next day to 5-6 days later to another destination in Japan. So as an example, last Summer on a Tuesday morning I sent my large suitcase via Takyuubin to Nagano from Kanazawa. I then used an overnight bag for my Tuesday night and Wednesday night in the Kamikochi area and had my big suitcase waiting for me in Nagano on Thursday when I arrived. I tend to travel with one large suitcase because I shop a lot.

Third, you can fly and just take your bag on the plane. JAL and ANA both seem to have pretty generous luggage policies and there are special airfares for foreign tourists.

Fourth, if you're on a tight budget many Highway buses have an underneath area for storing luggage if you are using one as your transport. (Local buses are awful with anything larger than a backpack though, only Highway and Airport buses have under bus luggage storage.)

So I am with @LikeBike. Go spend some time thinking about where you want to go and what you want to do and what is realistically within your budget. You have 5 weeks in Japan which gives you so many opportunities and options. I'm jealous!

Good luck!

Yep, luggage is no problem. @rkold gave you all the info you need.

I would love to go into the mountains, how hard is it to get there from Tokyo?
The closest mountains to Tokyo takes 50 min from Shinjuku: https://www.japan-guide.com/e/e3029.html (Takao-san)

But obviously there are many other mountains a bit but not terribly much farer away. And other many mountains much further afield. Japan is a very mountainous country.

On Meetup there is a very active hiking group in Tokyo. They go out hiking at least once a week, often even both Saturday and Sunday (and they speak English). https://www.meetup.com/Tokyo-Day-Trips-Meetup/events/260813174/
I went only once. I am not into hiking... cycling is for me...

Well, you certainly cannot bring in 10 huge suitcases per person on a Shinkansen, but then no other transportation system can.

I don't know how big your suitcase is, but a typical way to travel is to stay at a hotel in Tokyo at least for the first night and last night of your whole journey, leave your biggest luggage to the reception, and travel the country with a lighter backpack or the like.

If transportation fee isn't a problem, you can go anywhere. But since you don't seem to be used to traveling, I wouldn't recommend you to hike mountains or swim extensively in the sea alone.

You can come back to this forum after you've narrowed it down a bit more.

About the luggage, I will have one suitcase, limited by airline limits (max 20ish kg, limited by size too) and one hand bag (max 12kg, limited by size). ...That should be ok, right? Just making 100% sure.

Yes, I will be staying in Tokyo for first 2 weeks, then I plan to move elsewhere. No concrete destinations, as of now. Thinking about 3-6 places in 2 weeks?

Very well, I will think some more about the places and check the links.

As mentioned. No issues travelling with luggage.

So with that in mind. I'd just reboot your itinerary and start fresh. You can travel anywhere in Japan with you luggage.

If you are going in August September, where you go would be dictated a bit by how well you handle heat. It will be very hot that time of year, the further north you go the cooler it would be and also the less risk of weather issues from typhoons.