“And it shall be from new moon to new moon And from sabbath to sabbath, All mankind will come to bow down before Me,” says the Lord.
Though currently not explicitly defined as holy days, new moons are important for determining when the annual holy days occur. ("on the seventh day of the month" etc.).
This verse is describing life during the Millennium. It won't be only Israel that celebrates God's holy days, but all of mankind.
On every weekly sabbath and at every new moon, people will gather to worship God.
Weekly sabbaths are an obvious example of days that are holy to God (the fourth commandment), while a specific instance of an annual holy day is described explicitly in Zechariah 14:16:
And it shall come to pass, that every one that is left of all the nations which came against Jerusalem shall even go up from year to year to worship the King, the LORD of hosts, and to keep the feast of tabernacles.
Notice the "from year to year" expression. Like "from new moon to new moon" and "from sabbath to sabbath" it means every occurrence, not continuously all year long.