The short answer
If it is your first visit, you are travelling with family or older parents, you have heavy luggage, or you only have two or three days, a private car is usually easier than mixing taxis and the MRT. Singapore is compact, but the joins between attractions, the queues and the navigation still eat into a holiday.
Airport transfers
Send your flight number, arrival time, group size, luggage and hotel address to the charter company, then confirm the vehicle and quote. Arrival pickups usually include a reasonable waiting window after landing, and a good operator tracks delayed flights.
Planning a day trip
Two to four main stops a day is comfortable. For example, Marina Bay in the morning and the Botanic Gardens or Chinatown in the afternoon. Families with children should give Sentosa its own day and return to the hotel for a nap. Do not try to fit everything into one day.
Choosing a vehicle
Two to three guests with two or three bags usually need a sedan; four to six guests, a stroller or extra bags need an MPV; seven or more need a minibus; thirteen or more need a coach. If unsure, send a photo of your luggage.
What affects the price
Vehicle, hours, distance, area, waiting and parking all affect the quote. A good quote lists inclusions and exclusions clearly instead of giving one vague total.
One last thing
A private car should not be about filling every hour. Leave some space to sit down for a meal or look at the sea; it is often more memorable than one more attraction.