Is full wave rectifier better than half wave one?
Either can work correctly if designed properly. If you have a dumb rectifier supply feeding a 7805, then all the rectifier part needs to do is guarantee the minimum input voltage to the 7805 is met.
The problem is that such a power supply only charges up the input cap at the line cycle peaks, then the 7805 will drain it between the peaks. This means the cap needs to be big enough to still supply the minimum 7805 input voltage at the worst case current drain for the maximum time between the peaks.
The advantage of a full wave rectifier is that both the positive and negative peaks are used. This means the cap is charged up twice as often. Since the maximum time since the last peak is less, the cap can be less to support the same maximum current draw. The downside of a full wave rectifier is that it takes 4 diodes instead of 1, and one more diode drop of voltage is lost. Diodes are cheap and small, so most of the time a full wave rectifier makes more sense. Another way to make a full wave rectifier is with a center tapped transformer secondary. The center is connected to ground and there is one diode from each end to the raw positive supply. This full wave rectifies with only one diode drop in the path, but requires a heavier and more expensive transformer.
A advantage of a half wave rectifier is that one side of the AC input can be directly connected to the same ground as the DC output. That doesn't matter when the AC input is a transformer secondary, but it can be a issue if the AC is already ground-referenced.
Simplified explanation:
An ideal half-wave rectifier only "uses" half of the AC waveform (hence the name half-wave).
An ideal full-wave bridge rectifier will use the entire AC waveform.
An ideal full-wave rectifier (with a center-tapped transformer) will also use the entire AC waveform.
You can see that that for the half-wave rectifier, every second AC cycle is skipped leaving a gap in the output waveform. For the full-wave rectifier, since the whole waveform is used, the gap is gone (the effective output frequency is doubled).
If these waveforms are applied to a capacitor, you can see pretty clearly that for the half-wave rectifier, in order to maintain clean DC, the capacitor would need to be large enough to hold up the voltage during that big gap. For the full-wave rectifier, since there are more 'peaks', the capacitor can be smaller than for a half-wave rectifier at the same power level.
To your question, a properly-designed half-wave rectifier should have a sufficiently-large capacitor to maintain regulation despite only using half of the AC waveform, so the regulation should be just fine. There's no need to 'upgrade' the circuit with a bridge.
Just to clarify, the transformer doing the voltage stepdown works only on AC. The rectifier converts the AC to DC, which isn't to say that the voltage isn't varying, it's just that the current isn't going both ways. Full wave rectifiers are definitely more awesome than half wave rectifiers because they give you power during both halves of the cycle. They can even correct DC polarity reversal!