Instead of "transistor tester", search for "transistor curve tracer". There are a bunch of DIY methods out there, One of the simplest is use of a dual channel function generator and an oscilloscope. The siggen tenerates a stepped sawtooth and a triangle for each step of the first signal.
But maybe you should start with defining better what parameters of the transistors (and for what types of transistors) you want to test. You can test quite a lot with a power supply, a breadboard, two DMM's and a few passives.
On the top of the line you would have two SMU's, but that's not going to work out for your budget. And even with those you're still limited to pretty low frequency measurements. When you build the classical transistor amplifier with 4 resistors, you can test amplification on every frequency the rest of your equipment can manage.