CMOS on breadboard - Which P & N type MOSFETs to get started?
You can play around with making gates from transistors using the CD4007 chip. It contains two complimentary pairs plus an inverter. 



Some possible gates with one package:



The other thing that might be useful is a CD4016 quad transmission gate. With those two you could make just about anything, in theory. 

In practice, it would be very tedious to make even a 4-bit ALU using such low levels of integration. 

These days, most folks who are interested in making complex logic get into using FPGAs, which provide astronomical numbers of potential gates in a single package. There is a significant learning curve to use such parts.