What is the best way to serve AC power to a 12v - 5v DC regulator with 1Amp current?
Given the specifications you've presented, you need a transformer with dual secondaries: one at 14 to 16 volts center-tapped for the 5 volt supply, and one with about 24 volts center-tapped for the 12 volt supply. You have not specified the 12 volt current you need.

You cannot use a single secondary winding for both voltages. The rectified and filtered raw DC for the 12 volts must be a minimum of about 15 volts, to allow 3 volts across the 7812. This same 15 volts, when applied to the 7805, will dissipate 10 watts in the 7805 when supplying 1 amp, and I don't think you know how to build a proper heat sink to handle this.

To find a transformer, go to the web sites of suppliers such as Allied Electronics, Digikey or Newark, and use their search functions. Specifically, look at Power Transformers, and make sure you have specified both the correct input voltage range and frequency. You appear to be in India, so you need 230V @ 50 Hz.

As for transformer size, you take what you get. Transformer manufacturers are very tightly constrained by the magnetic properties of the core materials available, and if they could produce a smaller transformer for a given level of performance they would. After all, smaller is cheaper, so they could sell for less than their competitors and thereby sell more units, making a greater profit.

I very strongly suspect that you don't want to do this at all. What will give you better results, and cost less in the long run, is to buy a switching power supply with both 5 volt and 12 volt outputs. Not only will it be cheaper, it will be smaller.