The kids are old enough to do a little walking but not too much otherwise I can foresee complaints (I have a 6 year old ;-) )
I am sure they will love spending time on the beach and eating gelato (lots!) so make sure to plan for that. I just checked and La Spiaggia is in Monterosso and has easy access to the beach, so whether you want to or not you will have to spend your fair share there.
I would suggest getting the train card and seeing the 5 towns by train, not doing much hiking along the trails since you do want them to climb the many stairs to get to Corniglia ;-).
I suggest also trying out the boat bus that connects the bus, the best would be to get by train all the way to Riomaggiore and then take the boat back so that you get to see all towns from the sea, it provides different view and experience.
As far as budget, you will be visiting in the summer so don't expect eating out will be cheap, it is high tourist season --- make sure to ask your hotel hosts to provide recommendations since they likely have agreements for discounts at local places to eat at. I am not sure whether you will have access to a kitchen but it being just 2 days, you can stay on budget on other parts of the vacation.
Make sure to use appropriate footwear while there -- the towns are on cliffs and if it were to rain, the sidewalks and trails do get slippery. The CT are looking into limiting visitors in the summer and one way to ensure visitors don't endanger themselves is to prohibit unsafe footwear. Yes, there are tourists who try to go hiking with flip flops!! Dangerous and quite frankly, stupid. It is a sure way to injure yourself.