We are arriving on the 23rd of July and staying till the 18th of August... I think the Cheapest for me is to buy the Monthly card, But if I buy the Monthly card of August, I can start using it in the last week of July ( that is between 23 and 31 or I would have to buy one week for that too )... One more thing I have a 10 year old and a 12 year old are they both considered children so they get half price or not?
Navigo month card validity starts on the 1st day of the month. If you purchase a Navigo month card after 19th day of the current month (say July 20), it will only be valid starting next month, not the current month.
July 23 of 2019 falls on a Tuesday. A Navigo week card will be valid from Tuesday till Sunday (July 28), leaving 3 additional days before August (July 29, 30, 31, Monday-Wed). Those three days can be covered by Navigo day passes (minimum of 22.50€ for 3 days) or another Navigo week pass (22.80€). I'd recommend the Navigo week pass since it covers zones 1-5 (vs. 1-2 for the minimum cost day passes) and you do not have to renew the card each day like you would with the Navigo Day Pass.
After your initial July23-28 Tue-Sunday, you'll need transportation coverage from July 29 to Aug 18 (3 Monday to Sunday periods) it is actually cheaper to use 3 Navigo Week passes (3 x 22.80 = 68.40€) to cover that period than a Navigo week + Navigo Month pass (22.80 + 75.20 = 98€)
There's no Navigo discount for children unfortunately (besides on yearly passes for resident children), so they'll have to pay the full Navigo fare, same as any adult.