Travelling from Gare Du Nord to Gare De Lyon

I would order a G7 taxi the night before to pick you up at your hotel. With 4 people and little kids and luggage this is just the way to go IMHO. There is a fee to order ahead.

I dont mind travelling with the RER as long its busy and safe at that hour (because its early morning). Thats why i need to know if its ok to go with the RER.Thank you.

I always get the RER (line D). 2nd Stop, easy.
If you are already in Paris, make sure you have enough t+ tickets for the trip, saves 5-0 minutes at the machines. Train leaves from the underground platforms, No. 44.

If you have spent a few days in Paris, the métro/RER will be familiar and kids will be easier to "herd". It will probably be faster by RER than by car.
Video on how to do it here: https://www.youtube.com/watch?v=RSAHTKVBVwg

Edit: - Just read your second comment: "busy and safe at that hour (because its early morning)."
If that is a weekday, it will be the start of the rush hour, defiantly busy. RER will be the same as the métro, normal transport for normal people.
If the kids are old enough, get them involved in route planning and sign spotting. Get them to plot routes on the métro map and spot the signs for line numbers and station names.

Gare de Lyon is large and has two different parts. Leave enough time to find your platform.

RER-D is an easy and fast option.

Thanks Chris F. i think it is on Thursday then it must be busy. I've been to Paris few times and travel with kids. RER is the best way.

Thank you Nigel