The Altair 8800 is recognized as being the first home computer. It used an Intel 8080 processor, the forerunner to the 8086 later used in the IBM PC.
The initially available version with a 2K memory board cost $975, which would be about $3800 when adjusted for inflation. With that, you could toggle in a machine-code program in Octal on the front-panel switches, run it, and see the results in the lights. To do anything more required a backplane expansion and I/O cards which were not available until months later.