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You can go through a 'proxy server'. A proxy is an intermediate site which bounces all your browsing requests through effectively an IP masker. The sites you visit get the IP address of the proxy rather than your own IP address.
Proxy servers range from free and slow to paid and fast.
Yes, of course the proxy server has to know your IP address, or as you say you would never get any incomming data. But proxies generally have a policy not to log any traffic or visitors.
It is completely impossible to connect to the internet without anything knowing your IP address as in the first place your ISP has to assign an address to you, all you can do is to hide your IP address to sites beyond the proxy server you connect to.