We just found out that my daughter will be an AuPair in Madrid this summer. We already had a family trip planned to Amsterdam, Bruges and Paris which will begin about 7 days after she finishes up in Madrid. I am going to fly to Madrid to meet up with her and want to do some fun things before we connect with the rest of the family in Amsterdam. Should we focus our time in Madrid for those 7 days and fly directly from Madrid to Amsterdam or are there any suggestions on cities or countries to see along the way to make better use of our time? I'm open to any suggestions.
You could do day trips from Madrid to Toledo and then one to Segovia and El Escorial.
Yes it would be better to fly Madrid to Amsterdam
City halfway between Madrid and Amsterdam? Paris
You could spend a few nights there. Fly or train Madrid to Paris, train Paris to Amsterdam.
For me the decision would depend on whether your daughter will have been able to see much of Madrid and the obvious side-trip cities of Toledo and Segovia during her time off (limited though it may be). She might prefer to see a different part of Spain, in which case I'd pre-purchase AVE tickets to Barcelona (to get a really good price--but be sure you get the date and time dead right, as the promo tickets do not allow changes or cancellations). Six days in Barcelona would be great, and it will probably not be quite as hot there as in Madrid at the end of the summer. The train will take less than 3 hours.
When it's time to head north, you can easily fly from Barcelona to Amsterdam. That's another ticket to buy soon for the lowest price. Be careful of the departure airport. RyanAir sometimes uses Girona. Girona is a very nice place (worth going on a day-trip or spending 1 or 2 nights there, for sure), but if you're staying in Barcelona, you don't want to have to get to the Girona airport for a flight.
I'd spend the time in Spain. Toledo obviously, but also Cordoba. The Mezquita is for me the most impressive sight in Spain and I have seen the Alhambra which is also stunning.
https://janettravels.wordpress.com/2013/09/27/the-mezquita-at-least-the-catholics-had-the-good-sense-not-to-destroy-it-when-they-desecrated-it/
Spend a couple of days having her show you around Madrid and do some side trips to catch Toledo, Segovia, Cordoba at least.
If you felt that you have seen a lot of Spain then Paris would be a good interim stop -- but it sounds like you haven't spent time in Spain -- I'd make the most of that and presumably you have an interpreter to show you around in your daughter.