Learn the local emergency numbers on your first day, the same way you note your address; put them in your phone contacts, because in a real emergency you won't want to be searching. Countries vary: some use a single unified number for police, fire, and ambulance, while others maintain separate numbers for each service, and a fair few tourist destinations also run a dedicated tourist police or tourist hotline with foreign-language staff, which is often the most useful number for non-life-threatening problems like theft, scams, or disputes.
English availability at general emergency lines ranges from reliable in major cities to unlikely in rural areas, so have workarounds ready: your address saved in the local script, a translation app, or a local person to hand the phone to. Hotels will call on your behalf, which is frequently fastest.
A few more numbers worth saving on arrival: your embassy's emergency line, your travel insurer's assistance number with your policy number, and the nearest international hospital.