You need to make sure that you inactivate RNAses. I have used Trizol reagent for RNA extraction. It gives a good yield of RNA. Besides this make sure you are following all the necessary precautions while extracting RNA, namely wearing gloves, working with solutions, tubes and microtips that are RNAse free. Always be quick in the extraction process and work with ice because RNA degrades at elevated temperatures and alkaline pH.
What Malcolm suggested, those are much-needed steps. My points here, what is your cell numbers which you are seeding or using for experiments. If you have seeded enough cells as per plate size and planning to isolate RNA directly without any kind of treatment or intervention, then 12 well plate size is a very good setup.
Even if you are doing any treatment or intervention with a decent number of cells then also 12 well size plate is good.
If your cell numbers are not sufficient then RNA may come less. Even RNA isolation from 24 well plates does not always give a good concentration of RNA.