If you want it to run from mains, you need a transformer and that big capacitor to smooth out the output after the voltage is rectified using the bridge rectifier (or diodes). This will give you some DC voltage above 5v.
Alternatively, you can make everything run from a 9v battery or from 3-4 AAA / AA batteries. Naturally, as the batteries get depleted, the voltage slowly drops. The nice part about this is that you don't need big capacitors and rectifier, because the voltage coming from batteries is already DC.
So in both cases, you need a linear regulator to give you a stable voltage from whatever input you give it. Each linear regulator has a particular voltage drop, which is the minimum voltage ABOVE 5v that needs to preset at the input for the regulator to give stable 5v output.
This value is listed in datasheets but keep in mind this value is often specified for a current draw close to the maximum current the regulator will output, for example 2v at 1A. If your projects only require 100-250mA (and I think that's the case here), the regulator will most likely be stable even with less than 2v reserve.
With the option to power it from mains using a transformer, you can use any basic linear regulator with a voltage drop as high as 2v (therefore demanding at least 7v to output 5v) and simply size the transformer and capacitor so that the regulator will always see that 7v.
With the 9v battery, same story. The 9v battery will die at about 6.5-7v so you don't really have to take too big precautions - a regulator that needs 1.5-2v voltage above 5v will work.
A 9v battery however has little capacity, usually about 150-200mAh, so depending on how much you use those gadgets, you'll run through batteries like crazy.
It's more cost efficient to use 4 AAA/AA batteries, because these usually have 1500-2000mAh capacity. But, you have to pay attention to the voltage. Rechargeable aa/aaa batteries have about 1.35v when charged, 1.15v when discharged... 4 of them are too close to 5v to get a regulator to output 5v.
However, if you go with non-rechargeable batteries, those have about 1.65v each when full and about 1.35-1.4v when discharged, which gives you about 0.6v above 5v so a regulator with less than 0.5-0.6v would work fine.
http://uk.farnell.com/texas-instruments/lp2950-50lpre3/ic-v-reg-ldo-5-0v-to-92-3-2950/dp/1262363 up to 100mA, up to 0.38v drop,
http://uk.farnell.com/on-semiconductor/lm2931dt-5-0g/ldo-reg-37vin-0-1a-5v-3dpak/dp/2102542 up to 100mA, up to 0.16v drop
http://uk.farnell.com/infineon/ifx25001tf-v50/ic-ldo-reg-5v-400ma-to252-3/dp/2215565 400ma, 0.25 v
etc etc