Hi Miss Blueberry,
You need a lot of frequent flyer miles to get 4 people to Europe. The exact number depends on the airline program, but you are looking at approximately 240,000 needed. Luckily, you are planning well ahead of time so there is plenty of time to get credit cards for signup bonuses. I am going to suggest a two credit cards you might want to consider getting. Given that Denver is a United hub and American Airlines/US Airways awards to London typically have large fuel surcharges because you are on British Airways flights, I am going to suggest cards which can be used on United and other Star Alliance airlines. (If the last sentence sounded like gibberish to you, just ignore it).
1) Chase Sapphire Preferred
This card has a signup bonus of 40,000 miles plus an extra 5,000 if you add someone (ex. spouse) as an authorized user. For someone who was only getting one card, I would suggest this as you can use the points in a variety of different ways (on multiple airlines including United, hotel chains). This card has the annual fee waived for the first year.
2) Barclays Lufthansa Mastercard
This card has a signup bonus of 50,000 miles. Given that roundtrips from the US to Europe are 50,000 miles on Lufthansa's award chart, this is great. Note that you don't just use this on Lufthansa flights. You can use it on any Star Alliance airline (United, Brussels, Polish, Swiss, Air Canada). This card does have an annual fee of $79 that is not waived for the first year, but it is well worth it.
If you and your spouse each got both of these cards, then you would have two round trips from the Lufthansa cards and 1.5 roundtrips from the Chase cards (these points can be combined between spouses). You could definitely get another card to cover the last oneway, but I don't want to overload you with a bunch of cards.
Also, if you have a business (or any kind of sidegig that you can reasonable claim to want to keep expenses separate for), then you may also want to consider getting business credit cards. I can suggest some of these as well, so message me if you are interested.