The purpose of the Asset account is to track the real estate basis value, for depreciation purposes (as well as resale). So I would only put improvements in the Asset account that increase the basis value of the property which will then effect the depreciation schedule.

The treatment of appliances is a matter of preference. I would consider the purchase of the property including all appliances as part of the basis and depreciate the property. If you wanted to get more complicated, you can split off the appliances as their own assets, and lower the basis of the property accordingly. This will lower the basis of the property, which might increase capital gains taxes at resale. The benefit would be deducting the cost of the appliances as a write off the first year, since the appliances would be below the cap limit for fixed assets (typically $2500 ea).