Devilmaster, a transformer will not help in gaining energy. On the contrary, it will loose some electrical energy and produce some heat.

What it will do is (depending on how it is used) either reduce amperes and increase voltage or viceversa - except for the heat loss the product of amperes and voltage is the same on both ends of a transformer.

And instead of using such a complicated mechanism as you suggest (connect the wheels to something that spins), it would be much easier to use a bicycle hub dynamo and build the hamster treadmill from that - this will reduce mechanical loss of energy (friction --> heat) to a minimum. If the treadmill has a diameter the same as a bicycle wheel (28") the weight of a hamster is enough to start the thing turning - I know because I got such a bike hub dynamo in my little bicycle.

And if you use instead of lightbulbs some of the new super-bright LEDs you can get quite a lot of light out of a little hamster.