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I'm interested in how they used numbers from 1 to 10 in Japanese today?
1 ichi, 2 ni, 3 san, 4 shi, 5 go, 6 roku 7 shichi, 8 hachi, 9 kyuu, 10 juu.
or
1 ichi, 2 ni, 3 san, 4 yon, 5 go, 6 roku 7 shichi, 8 nana, 9 kyuu, 10 juu.

Thank you!

IIRC, they're both used, and it's just that shi and yon are used in different contexts (plus shi is popularly associated with death, since it sounds like some words having to do with death, and so 4 is widely seen as a lucky number and perhaps people avoid using that particular form as well).

Hachi is a Sino-Japanese word (so basically, a loanword from Chinese); nana is a native Japanese word.

And nana is 7.

When numbers are used independently, such as when counting or doing math, shi and shichi are generally used. In quick counting and various set mathematical expressions, kyuu can be reduced to ku. For phone numbers, serial numbers and what have you, yon and nana are typical because they are more audibly distinct. When numbers are used with counter words in order to count specific nouns, yo(n) and nana are standard. With counter words, the other native Japanese numbers occasionally show up, but these never occur independently:
1 hito-
2 futa-
3 mi-
4 yon-
5 itsu-
6 mu-
7 nana-
8 ya-
9 kokono-
10 too-

Janko Gorenc wrote: 1 ichi, 2 ni, 3 san, 4 shi, 5 go, 6 roku, 7 shichi, 8 hachi, 9 kyuu, 10 juu.
or
1 ichi, 2 ni, 3 san, 4 yon, 5 go, 6 roku, 7 nana, 8 hachi, 9 kyuu, 10 juu.

There was a mistake. (I just want to make sure you understand it)

shichi sounds like ichi, so becomes nana
shi sounds like "death", so becomes yon

yon and nana are native roots, the others are chinese roots.

clawgrip gave the native roots. for example as counter words these become:
1 hitotsu
2 futatsu
3 mittsu
4 yottsu
5 itsutsu
6 muttsu
7 nanatsu
8 yattsu
9 kokonotsu
10 too

or with dates they are:
1st tsuitachi (*a different word)
2nd futsuka
3rd mikka
4th yokka
5th itsuka
6th muika
7th nanoka
8th youka
9th kokonoka
10th tooka
(11th+ mostly use the chinese roots)

when counting, you can use either system (ichi-ni-san-shi OR ichi-ni-san-yon)

but the second system (ichi-ni-san-yon) is more common

when it's used with another word, they are sometimes set. the names of the months in japanese are based on numbers + -gatsu (month) - in this case, April is shigatsu, not *yongatsu, and July is shichigatsu, not *nanagatsu. as clawgrip said, September is kugatsu, not *kyuugatsu (this form changes too).

hope this helps!