Travel around Kyoto

I am first time visiting Kyoto. I notice there are places to visit in Central / Eastern /Northern Kyoto.
Is there a train system we can use to travel around Koyto to most of these places of interest?
Is it ok to stay in a hotel near to train system and not having to change the hotel during my 6 days in Kyoto?
Is there other accommodation other than hotel?

There are good places to visit all over Kyoto.
By "train" I take it you would also include the subway. Ultimately it's all up to where you plan to go, but unlike many other Japanese cities, rail is only of limited value. You can of course take the bus, or rent a bicycle if you're not carrying a lot, or even a taxi if you have the money to burn.

There are often several other good places around someplace you choose to go, so it makes more sense to get there however you want and then walk around the rest of the way. Avoid zig-zagging all over the city to use your time better.

Is there a train system we can use to travel around Koyto to most of these places of interest?
Kyoto has a huge bus system that you can use to get to almost all the sites you want to see. There is also an underwroud subway system. Please look under transportation in Kyoto for more detail. This site has a lot of good planning information. For example -
The JR rail will help you going from Kyoto station to Arashimaya, but you could also take a bus or a taxi.
https://www.japan-guide.com/e/e2363.html

Is it ok to stay in a hotel near to train system and not having to change the hotel during my 6 days in Kyoto?
Yes, if that is what you wish. I have found it hard to switch hotels in the same city because you loose so much time in packing and switching which could be used site seeing. As mentioned before it is up to you. I know my first trip I switched hotels a lot to experience all the different lodging choices (which let me try a little of everything but got tiring).

Is there other accommodation other than hotel?
Yes, once again look at the site
https://www.japan-guide.com/e/e2025.html
I personally enjoyed staying at Hana Hostel in Kyoto. It is close to the station, staff is very helpful and I got to sleep in a Japanese room.

This site might be helpful for your question.
https://www.booking.com/guest-house/city/jp/kyoto.html?aid=1200854&lab...

If you are packed lightly, switching hotels isn't that bad, my first time in Kyoto this November and I will be staying at two hotels, apparently the price for two night at the same hotel, turns out really expensive, one night was about 5000yen, the other was 8000yen, so I found a nearby hostel with a private room for 2520yen.
so now its 5000 yen + 2520 yen.

To travel around Kyoto, you probably want a mix of train, subway and bus, as someone said you can rent a bicycle, and I have a thread opened about that which is only a day or two old, if you need information on that.

Seriously, for 6 nights I'd stay in two hotels...at different ends of the city.

Split your sightseeing into two groups, based on where you're staying.

Enjoy some variety...

I've been to Kyoto a few times. It is compact and very easy to get around.

The subway is excellent, but you may need to take a bus as well, depending on where you are going.

Get a bus pass from your hotel at JPY 500 per day and is cheap transport.

Get Pasmo or an ICOCA IC travel card, to save hassles when buying subway or bus tickets.

Switching hotels would take more time that than the few extra minutes of travel at the beginning and end of each day, then you have to learn the new local layout as well.