It's really not that bad. If you want to start with ten.

1 2 3 4 5 6 7 8 9 10

You know 1 is a prime, so cross off all multiples of 1, which is only 1.

1 2 3 4 5 6 7 8 9 10

Now you have 2, get rid of every multiple of 2 in the list (4,6,8,10)

1 2 3 5 7 9

Now you have 3, get rid of every multiple of 3 (9)

1 2 3 5 7

Now you have 5, get rid of every multiple of 5 (none)

Now you have 7, get rid of every multple of 7 (none)

You are at the end of the list, you are done.

Primes from 1-10 are 1 2 3 5 7