Though Wishes can be bought with real money, you'll also earn them at a healthy pace just by playing Genshin Impact. The two currencies that can buy them are Prismogems, which you'll earn for doing basically anything like fighting bosses or doing missions, and Stardust, which you earn for opening Wishes. An in-game shop will tempt you to spend those currencies on a variety of items, but don't fall to temptation: Save your Prismogems and Stardust to buy more wishes.
Prismogems can be converted to Wishes directly in the Wish menu, but Stardust has to be converted using the shop menu, in the section called "Paimon's Bargains." You can only buy five of each type of Wish a month using Stardust, though.