I am currently able to restrict places to only one country, but I also want to restrict on a specific city also. Any ideas how to achieve that? This is my current code:
Places Autocomplete currently allows you to use componentRestrictions to filter by country. What I would do in your situation is to use the options argument to filter by the bounds that define the city in question:
var cityBounds = new google.maps.LatLngBounds(
new google.maps.LatLng(25.341233, 68.289986),
new google.maps.LatLng(25.450715, 68.428345));
var options = {
bounds: cityBounds,
types: ['geocode'],
componentRestrictions: {country: 'est'}
};