It would be common to use "the" in those contexts.
"I'm going to the post office" or "to the gym" are perfectly correct.
"I'm going to the shops" would be more common (as you would probably be visiting multiple shops for the particular purpose of shopping". But if you doing a particular type of shopping, you could say "I'm going to the supermarket" or "the hardware store", for example.
So:
- The shops There is usually only one collection of shops in a town. You say "the shops" because they are the shops you usually go to, for the particular purpose of shopping.
- The pub. Many people will habitually visit one pub, even if there are more than one pub in the town.
- The bank. *Again, a town may have several banks, but I only use one of them.
- The supermarket. Many towns only have one, even if there are several, I'll normally only visit one of them regularly.
- The cinema Only the largest cities have more than one cinema.
But if the reference isn't clear you'd be less likely to use the.
- The petrol station.
- The shop *Contrast this with "the shops".
- The house
- The cafe
If from your personal context any of these could be specifically understood you can use "the". If a village only has one shop then "the shop" is understood. If you and a friend meet at a particular cafe after work each day then "the cafe" is understood. If you are talking to your housemate and you talk about "the house" the identity is understood.
Now suppose you are at a petrol station, and there are several in the town, all of which you use equally, and your friend asks you where you are. You don't say "I'm at the petrol station." Since that doesn't answer the question. You might say "I'm at the petrol station on Oak street", or "The petrol station by the supermarket". By saying "by the supermarket" you identify the station, so you use "the petrol station". Or you cooperatively understand that your friend doesn't really want to know where you are but why you aren't with them, so you might reply "I'm getting petrol".
The heart pumps blood around the body
. Why is there "the" in "heart" and "body"?