Hello, it's me and my two kids first time flying and wanted to know if can we bring an Cary on bag on plane with purchased tickets?

You can not bring a carry on bag free with every airline, so you have to check the airline you purchased tickets from

Depends on airline

Yes you can bring a carry-on.

You can, as far you follow the dimensions & weight airline's rules. Check your flight reservation.

I suggest you look at your booking information. All airlines and ticket types have different rules.