Simple example of double sided PCB design

What exactly is your problem?
- how to use pcb layout software?
- how to make a double sided pcb?

There is no special or explicitly simple circuit for either single sided or double sided pcb layout. You can make a double sided layout from almost any design (provided it is not too complex for 2 layers which means it requires more than 2 layer - that is of no concern here).
You could start with a single resistor connected to two terminals, one connection using the top layer, the other connection using the bottom layer. This I would consider a most simple "design" although it is rather meaningless. A circuit that is also rather simple but at least creates a little effect is an astable multivibrator, e.g. this one (used to be kind of a "hello world" project for aspiring electronics hobbyists). You have crossing connections here which can easily be made using two sides of a pcb (although it wouldn't be strictly required here, but you're looking for an exercise).

 

It doesn't have to. Point is the op wants to get started with double sided pcb design and asked for a simple circuit...

I looked at it. Seems o.k., Now increase the level of difficulty with a more complex circuit. In the end the proof lies in the fabricated pcb: does it work or nor?

 

You can make a tryce alternating between both sides of the pcb as often as you like using vias.

In my opinion your approach to double sided pcb design is not very useful. Instead of trying to route a simple circuit on two layers (which is oftne possible on a single layer for simple circuits), find a moderately complex circuit that may even have some use to you and make a layout for it. Then have a pcb made and check whether the final product works as expected.

You may also want to look at publicly available layouts and tutorials like e.g. this one.