Should I buy breadboard with power supply?

Whatever you do, you will need a power supply of some sort - having it built into the breadboard may be convenient.

If you expect to work with microcontrollers or other digital logic, you will probably want a 5 volt supply (although many parts these days are designed to work on lower voltages.)

If you will be working in the analog world (audio amplifiers, for example) you will want bipolar supplies +/- 12 - 15 volts, typically.

As a general rule, the breadboard + power supply kits don't save you that much and are a lot less flexible than separates. In addition, you may want something other than +5 if you ever use op amps or lower voltage stuff. Something like this

It would help, though, if you could mention some specifics about what equipment you have in mind.

"stuff" (cause I don't know) with Microcontrollers and Microprocessors.

Just get one without built-in supply, then you can experiment whatever you are building *even* on stability with respect to power supply Also, depending on what you are trying, more as one supply is needed so put money in a decent dual output supply of say 6...10A 0..30V

Thank you.

I'd get one without (it's more flexible) and use some of the money saved to buy a few 7805's. Personally, I mostly use a 9v battery with a 7805 and capacitor soldered directly to the battery clip - effectively a regulated 5v battery :-)

So I don't have to have one plugged into the wall?

------------ No, any battery pack greater than about 7V can be used with a 7805 voltage regulator to provide regulated +5.0V +/- 5% .

-Steve