Here's a comparison of the inflation rate for the CPI less food and energy, and the inflation rate for food and for energy. As you can see, rising energy prices by themselves don't seem to lead to higher inflation down the road. Ditto food prices. They will however make the overall CPI more volatile than otherwise. Their noise, statistically, so eliminating them doesn't change the bottom line while making the bottom line easier to see. Go to the BLS charts page and click on CPI vs CPI less food and inflation to see the difference.
Here is "core" inflation (the CPI less food and energy) versus (i) healthcare services, (ii) new car prices and (iii) apparel prices. On average the latter two have been flat (+0.1% for cars) or falling (-0.3% for clothing). Old houses didn't have closets. New ones have walk-in closets. In contrast over the past 20 years overall "core" inflation has been 2.0% ("headline" everything included inflation has been 2.2%). Healthcare has averaged 3.8%. That means that over the past 20 years "core" prices rose 50%, while healthcare prices more than doubled [compounded 20 times = (1.02)^20 vs (1.038)^20 in Excel]. That makes sense: out of the past 241 months, the prices for healthcare services rose faster than core inflation in all but 4 months.