I am looking for the most famous alternative inflation metrics used by some acknowledged economists other then CPI and PCE. In my opinion energy, food, education, healthcare, housing and transport is much more important then say a cellphone or a new tv, if anyone knows of any index weighting in this manner and have a similar idea around why to weight like this, please let me know.
CPI and PCE have already taken into account the "importance" of different goods and services based on how much an average household spend for items in those categories. You could look at the weights here.
Your intuition seems to be correct. Food and beverages take up 14%, Housing 42%, Transportation 16%, and Medical care 9%. These four main categories take up more than 80% of household expenditure (Education is about 3%), while Telephone hardware accounts for only 0.072%.
The PPI (Producer Price Index) measures the domestic output of raw goods and services.
The GDP deflator measures the changes in all goods and services produced in an economy.
For your purpose the US Bureau does release indexes that drill down further into the PPI in the form of industry level classifications and commodity classifications.