There are several ways to do that, and all of them involve credit cards. As you know, there are credit cards that give you rewards. Such cards can significantly reduce the price of your plane ticket or a hotel stay. When you book, you get some percentage back in the manner of points or miles. Those money you can use again for award flights or hotel nights.
Expedia even has its co-branded card that could help you earn points when you buy or book with Expedia. The Expedia®+ Card from Citi (expired offer) is quite useful for those who use Expedia a lot, as it offers 3 points for Expedia purchases. You can add to your earnings if you spend $1,000 in purchases within 3 months of account opening - that's quite moderate in terms of most cards, although the sign-up bonus itself is more than modest - only 15,000 points. Points are redeemable for Expedia purchases, for example, flights and hotel stays, so that may be suitable for you.
However, there's a way to knock down your costs even more - if you apply for a general card. All in all, general cards are more profitable and easier to maintain in the long run. Travel credit cards will give you miles or points that are just as redeemable for Expedia purchases. And actually you there are will be more miles and points to earn. Travel credit cards have considerably larger sign-up bonuses and the accrual goes faster.
One of the best ways to rack up miles fast is to obtain the Discover it® Miles. This is a super effective card for travelers. 1.5 miles on all purchases will let you earn even on your grocery and dental procedures. Even if you book though Expedia frequently, it's hardly your primary item of expenditure, so Discover's 1.5 miles will eventually be more than 3 miles for Expedia only. Besides, Discover matches all your first-year miles at the end of the year (provided you are a new cardmember), which is a kind of a welcome bonus, and unless you spend less than 15,000 on the card, it's going to be larger than on Expedia.
Note that Discover's miles are redeemable on Expedia and on other purchases as well. Miles never expire and you can earn any number of them. Discover has no annual fee, good to know when you don't want to overpay.
And there's also one thing that makes Discover superior - it has 0% intro APR on all purchases for 14 months (when the introductory interest rate is over, the ongoing APR will apply). Expedia cards have high ongoing APR, so you won't want to mess with carrying balance. Discover, on the other hand, lets you book now, pay later, even purchase a whole vacation.
In truth, it's your decision how to best save money on Expedia bookings, but you can consider some things suggested and choose a card to apply for.