For a first try, as long as you don't have really challenging problems and some experience, I'd vote for a cheap model.
If you really encounter speed problems (with spikes,,,) then you are better off with a scope. But for testing serial data transmission or sequences of output signal changes even the slowest logic analyzer will be sufficient with a 16MHz Arduino.
Stand alone analyzer or scope are for quite special cases or field service persons. For now computer memory and display should be sufficient and easy to handle.