What NAND gate chips should I use to create a simple computer
a CMOS part like CD4011 ot 74HCT00 is a good choice, they are reasonably energy efficient  which important if you're running several hundered chips.  maximum clock speed won't be spectacular, but should be enough to run demosnstation programs.
if you assemble the CPU in parts and test each part you should eventually have a working machine.