These days, some roads a computer hobby would take include:
Software - Windows, MAC, Linux, Web/Internet programming, build your own OS, build your own programming language; program a major platform in assembly language
Microcontrollers - Microchip PIC, Atmel AVR, various ARM processors, others; with microcontrollers, you can explore a number of hardware topics that modern workhorse OS's (such as Windows, MAC, and Linux) get in the way of
Programmable logic - hardware: FPGA, CPLD; software: Verilog, VHDL; build your own CPU; resurrect old CPU's, gaming platforms, microprocessor development kits; do things with FPGA or CPLD that would normally be done with microcontrollers
Hardware interfacing - build your own devices (typically with microcontrollers) to communicate with a gaming platform or cellphone
Unless all you're interested in is software on current major platforms (or old dead ones), you will need to do some electronic design work.