In Indonesia, reducing fuel subsidy strongly impacts the price of everyday goods. Why? In Indonesia, petrol/gasoline price is regulated by the government and historically it is heavily subsidized. I know, economists typically consider fuel subsidy to be a bad policy, for many reasons, including: it appears to disproportionately favor the wealthy who have more cars.

However, in Indonesia, fuel price seems to strongly affects other prices in general. I mean, when the government tries to reduce subsidy and increase the price, we will see prices of everyday stuff like food increase significantly. So, some people say increasing fuel price actually affect the poor the worst, because they barely have enough money to buy basic needs. The effect is so bad that such attempts to reduce subsidy are usually met with big protests by the general population. 

Some source:


Jakarta Calm After Subsidized Fuel Prices Are Increased
Indonesia inflation jumps after fuel price hike
INDONESIA: Poor hit hard as fuel prices rise.


I've lived in other countries where fuel price is determined by market and fluctuate freely (like say the US), but I don't see such phenomenon (e.g. grocery prices don't noticeably increase as much when gas price increase). Why?
Consumers rarely have to pay the full price of petrol / gasoline. That's true in most of the world, and the US is no exception. There are explicit subsidies and favourable tax regimes in many extraction countries; and of course there are the implicit subsidies in the unpriced pollution - oil's externalities.

Those explicit and implicit subsidies have been there for decades: and consequently, oil products are hugely over-used, compared to the alternatives: that's a direct consequence of oil being a product with demand that's not perfectly inelastic - indeed, estimates of long-term price elasticity of demand for petrol are of a figure around -0.6 or so. That means that oil is embedded in large quantities in the supply chains for very many products.

The larger the historic subsidies, the bigger the impact of changes in subsidy levels to the price of other goods and services. And the poorer the population, the bigger the impact. That's why you see a bigger impact in Indonesia than in the US.

Now, although in the long run removing some of the subsidies will correct some of those imbalances, that will require capital reallocation and some industrial restructuring, all of which takes time.

And although in the long run, rising prices of some products, in the absence of rising wages, will choke off demand for other products, leaving inflation at no net change, the short-term effect of rising oil prices is to raise the costs in a significant part of the supply chain for a lot of goods and services, given an instant boost to inflation.