What should I wash my plane with?
You should wash your aircraft in accordance with the manufacturer's instructions.

We could all offer opinions and product recommendations in response to this question. However, the best answer is: in accordance with the manufacturer's instructions. Since you ask the question generically, without any information on what aircraft you might have, I will answer generically.

Your manufacturer's instructions can generally be found in at least two places: the owner's manual/AFM and the maintenance manual. For example, in the owners manual for a 1975 Cessna 172M, you will find Section V: Care of The Airplane containing instructions on how to wash painted surfaces, bare aluminum surfaces, interior surfaces, and the windscreen and windows. The maintenance manual for the same aircraft contains nearly—if not exactly—identical information in chapter 2.

Now, for that aircraft and others, Cessna recommends use of mild detergent in certain applications; you might not find that overly helpful in choosing what to use. Based on my research and experience, and time honored practice that I've observed, Dawn dish soap is widely used for cleaning painted Cessna aircraft with no apparent long term harmful effects. The airframe that I fly for work has tens of thousands of hours on it, and while I cannot vouch for how often it was washed in the past, I try to wash parts of it a few times a week. There is no apparent washing damage to the paint.

I will also highlight Cessna's instruction to use Stoddard solvent on areas of stubborn oil or grease; this can be especially helpful in cleaning the belly with it's typical collection of oil and dirt. My practice is typically to wipe down the belly with solvent first, and then follow with a water and detergent wash.

If the manufacturer's instructions are not clear, or are otherwise insufficient to answer your question regarding your specific aircraft, I would recommend asking your maintenance professional. He or she should have both the knowledge and experience to steer you in the right direction.

For more insight into the world of aircraft cleaning, and especially for the specific product recommendations you want, you might take a look at this AvWeb article by Kim Santerre: Aircraft Cleaning.