Power Supplies are quite an issue and a BIG subject.
There is little point going into great detail here as this thread will probably be 'lost' in a week's time.
First, you have to determine the voltages and loads required.
Some projects may require several voltages e.g. a 5V for PICAXE and 12V for a motor or relay.
If we mix in to the equation Newbieness and Budget restrictions then we have to be careful. AlWAYS check the power rating. ALWAYS check o/p voltage. Trust nothing - especially if not a well-known brand.
Wallwarts (mains plug-in AC Adaptors).
You can get good or bad.
You can get regulated or unregulated.
You can get linear or switched.
PC PSUs (and other 'liberated' PSUs).
Many tend to be switched-mode and most will easily give out enough oomph to pop an unsuitable circuit. Some need min. load as mentioned. Don't use them in ignorance of spec.
Remember, all the above are usually designed for use with other circuits.
In many cases, with Switched Mode desgins, the final bit of smoothing is probably done on the target PCB.
So, filtering/decoupling/bypassing capacitors will almost certainly be needed on your PCB.
Ooomph!
Most of these PSU will be able to provide plenty of power/current.
Some will have lousy regulation and may be electrically noisy.
(Get an AM radio and put it next to it).
This also means that a cocked-up circuit can easily go POP!
Use a fuse or lose.
Voltage.
Beware.
Always check - regardless of the label. Sometimes there may be a fault.
Or there may be an error in the translation from Chinese.
Making your own.
If you decide to make your own regulator, say from 12V to 5V be aware of heat and power.
If using a LINEAR (normal/ old-fashioned) type then DO the POWER CALCS to see if you need heatsinking.
And READ the F. DATA SHEET to see what capacitors are required for YOUR DEVICE.
Note.
If your supply voltage to your regulator (e.g. 12V) is noisy and spikey a 5V regulator will NOT stop the noise. HF will go through it and into your circuit. So check with 'cope as you may need supplementary filetring LCR.
If using Switched Mode (e.g. Simple Switcher etc.) then READ the DATA SHEET to see the recommended components.
NOTE: With Switchers component choice is crucial for top performance - you may even (I'm SO SORRY) have to get out your calculator to work things out.
ALSO with Switchers your PCB track layout/design can make a huge difference to performance. A breadboarded Switcher will be poor compared to the well-designed on-PCB equivalent.
Top design requires a good ( I mean good) 'scope. Not a £10 USB plug-in.
Again, be prepared to protect your PICAXE circuit with a fuse.
The clever project-builders could build a trip with an Op-amp and transistors.
Home-made switchers.
With MOSFETs and drivers. Newbie.... forget it. Getting something really good (Lab spec.) is way beyond 99.99999% of people. Something good enough for a battery charger is possible but I'm sick to death of MOSFETs.
The BEST?
For prototyping you CANNOT beat a proper bench PSU with Current limiting.
The best Bench PSUs include Thurlby-Thandar (TTi) , METRIX , ELEKTRO AUTOMATIK , HAMEG ('n'chips). AGILENT.
The best ones use opto knob adjustment (no dirty pots) and some have serial control inputs.
None will be cheap, but top quality stuff will usually last longer than the operator.
And Current Limiting is faster than fusing and will save you money ( replacement popped components/PICAXEs inc Postage).
Yes, they empty your Piggy Bank. But as Christmas is coming the above should be able to justify it to Mum and Dad.
Cheaper options are available but I can't vouch for quality or durability.