You are perfect positioned to do this entire trip for whatever you want it to be on a sliding scale of full price to free. Just depends on how many cards you are comfortable with. I’ll start by saying that applying for cards will only affect your score in the very short run (2-4 months) and if you are good about making all your payments, it will pass your current credit score for the better.
Assuming you are comfortable with applying for multiple cards, here is the strategy I’d suggest.

Now:

Your wife:
1) Chase Sapphire Reserve (100K UR points, $450 fee, $600 travel credit)
2) Chase Ink Preferred (80K UR points, $95 fee)
You:
1) Chase Freedom (15K UR points). This is because you don’t have any other cards and this is a good starter card. Hard to get approved for anything else at the moment.
Once you meet minimum spending on each, you are at over 200K UR points. Flights to Spain in Economy will be 120K total round trip with United points (UR partner). And you can add in another destination for free within Europe. Book flights now.

3 months from now:

Wife:
1) Chase Hyatt (2 free nights at any hyatt in the world, $75 fee, 1 free night a year)
2) Chase United (50K united points)
You:
1) Chase SW Plus (50K SW)
2) Chase SW Premier (50K SW)
Doesn’t help for Spain trip but At 110K SW points, you can designate a companion (your wife hopefully) and you get free flights on SW for 2 years.

6 months from now:

Wife:
1) Chase Marriott (upto 100K points)
2) Citi Hilton Reserve (2 free nights)
You:
1) Chase Sapphire Reserve (100K)
2) Chase Ink Preferred (80K)

Now you have 315K UR leftover to use @ 1.5c per point towards hotels (or another trip), $1,200 in travel credit from 2 Sapphire reserve cards for AirBnB and stuff, 100K Marriott for Marriott Hotels, 2 Hyatt Free nights to use at the Park Hyatt Mallorca if you want to add that free leg on United and 2 Hilton nights.
If you want to keep going, at 9 months add hotel cards like Chase IHG, AmEx Hilton, Citi Hilton etc.

Tweak it as you wish. The reason I suggest 2 cards at once is because Chase will combine credit inquires made on the same day into 1. Good for your score. Suggesting Chase cards first because these are the best cards to get and Chase won't give you their best cards once you have over 5 new accounts within 2 years. 3 month intervals because it gives enough time for your credit score to bounce back.

You could also switch your wife's United app to now instead of Ink Business preferred and apply for the Ink in 3 months. This will free up 50K UR as you will just be using UA miles instead.

If you want to go all out on this one, you will have enough miles to fly in First class (like on Lufthansa).

Finally - I would totally disregard AmEx plat suggestions, with due respect.

This post was edited on 12/15/16 at 1:30 pm