You could use a mains transformer, say 120V primary, 10V secondary. Then use a bridge rectifier and large smoothing capacitor. This will give a fairly large amount of ripple at 100Hz / 120Hz, depending on your local mains frequency. Depending on the application, if you need better smoothing, a low drop-out regulator could be used. This will be the simplest solution. Don’t try making a direct off-line switched mode supply.. You could use a simple switched mode supply after the mains transformer to get the smoothing at a higher efficiency.
You have not indicated how well regulated the output must be. Is unregulated power OK or do you need 0.1% regukation or something in between. The poorer the regukation, the lower the price.
The other point is that this sort of thing is at the very basic end of one's technical education and if you need help on this, what are you doing that's more complex?
Third point - You can buy of the shelf 12V power supplies. Are you sure you can't use 12V instead?
I doubt that you can build this one as cheaply as you can buy it. Look through the pages of power supply manufacturers. Simple low-cost switchers will do your job and many have remote sense imputs that will allow you to adjust the output voltage from 12 to 10.
