Getting started with a microprocessor
What are your design requirements or goals? Any areas you expect to need to learn more about, or any areas you wish to focus on?

Because most of the Intel 8-bit microprocessors were so popular they are fairly easy to find as New Old Stock (NOS) or Pulled/Refurbished (reclaimed/recycled, usually tested). At least the 8080, 8088, 8086, and 8085 are available from distributors or obsolete / old-stock supplies. 

Two others that are worth considering would be the Zilog - Z80 (technical description), and MOS Technology 6502 (technical description). The Z80 was created by Federico Faggin, who designed Intel's 8080, and made the Z80 a binary compatible (but enhanced) processor.

The Z80 requires fewer support chips than the i8080 with built-in DRAM refresher, a single (5V) voltage supply requirement (i8080 required +/-5V and +12V), and can be run at any clock speed up to its specified speed (e.g. 2.5 MHz for the first NMOS models). 

Combined with 5V power supply, oscillator clock (e.g. TTL/CMOS oscillator "can"), Static RAM (rather than Dynamic RAM, DRAM) you only need a ROM/EPROM/EEPROM and interface (e.g. interface chips for parallel or serial interfaces) to built a minimal microcomputer. 

One fairly popular book on building your own microcomputer using the Z80 was Build Your Own Z80 Computer (available with copyright permission of author/publisher) by Steve Ciarcia. Also check out the (dated) alt.comp.hardware.homebuilt FAQ, and the N8VEM community for additional resources and references.

The MOS 6502 was the another popular microprocessor, in part because it was so much cheaper ($25 USD) than the Motorola 6800 which was originally $300 USD circa 1975. It was used in popular systems such as the Apple I from Apple Computers, Commodore KIM-1, PET, and Vic-20, BBC Micro, and the Commodore 64 and Atari 2600 game system used 6502 derivatives (6510 and 6507). So there is a lot of material available from retro-computing and retro-gaming people online, and parts. The 6502, like the Z80 was produced by several sources (i.e. second sourcing) including Rockwell in additional to the primary designer / manufacturer, MOS Technology.

If you have a particular (strong) interest in the x86 or IBM PC / XT history then a 8088 or 8086 might be a educational target to consider. Otherwise I would lean towards the Z80 as my first pick, and the 6502 as my second choice due to parts availability and resource material availability. 

The range of options is unlimited from a microcomputer built from almost exclusively discrete transistors, to a 25MHz 32-bit MC68030 workstation.