Another few points to add to the above:
Shoe Leather Costs: As inflation decreases the real value of cash, it means that people keep less cash on them. This results in more time spent walking to the banks/ATMs to withdraw money. This is an opportunity cost of wasted time.
If it is due to cost-push inflation, then it can kick off a price-wage spiral, where workers ask for higher wages to keep their real wages the same. These higher wages lead to further inflation, and so on.
Inflation hurts savers, as their savings depreciate in their real value. Conversely, it helps borrowers, so anyone in debt (note: government) benefits from inflation, as the real value of their debt depreciates.
If there is high inflation then hoarding can occur, this is when people buy durables as a means of keeping wealth (as currency is depreciating in real value). This can lead to shortages.
In times of high inflation, the price signal is lost, this means that there can be a loss of allocative efficiency as producers cannot respond to the signal of falling/rising prices as efficiently. Gluts and shortages of products can occur.
One positive point to note is that a stable rate of inflation can increase investment, leading to faster growth. This is because it disincentives those businesses from holding cash relative to investing in capital goods. Thus, to avoid inflation, businesses will increase their stock of capital.