Take the A Train. Or a bus.
I have visited 34 cities in 23 countries, some of them several times, and this far I have never used a taxi abroad. I travel mostly in large cities, so there is always a public transportation options available, also between the airport and the city. I also travel with reasonable luggage, mostly just backpack, so it is easy to move around and fit in to a bus for example.
Price is of course one reason to avoid taxis, as the price of a taxi ride is usually at least 10 times the price of a public transportation ticket. Public transportation is also more environmentally friendly than a car, and in some cases even the fastest option. No traffic jams in metro, just a bit tight sometimes. Still the main reason for me to avoid taxis is that for an introvert, it can sometimes be too intimate to be in a small space with a stranger. Unfortunately taxis are also a possible place for a traveller to get ripped off.
Observe daily life
In a bus, metro, tram, or a train it easy to blend in and also experience a part of the local daily life as an observer. One time in Tokyo metro I looked around and started to think about the other passengers, how we see each other only couple of minutes and then disappear to continue our lives. That could of course happen anywhere, but it was bit of an existentialistic moment!
Using public transportation forces you to be more aware of your surroundings and find out your route beforehand, which decreases the change of getting lost. And if you still end up in the wrong place, usually you can just go back and try again. Most cities have at least some kind or route planner in the public transportation provider’s website and Google route search works well too in many cities. You would think that you can’t get lost with a taxi, but have you watched the Amazing Race?
It pays off to do some research beforehand about different ticket types and where you can buy them. In some cities you need public transportation only occasionally, so it is cheapest to buy single tickets when needed, while in other places you will go everywhere by metro or tram, so for example a weekly pass or a 72h ticket is not only the cheapest, but also the most convenient option. For example in New York or Tokyo where you will use the metro several times a day, buying a single ticket every time makes no sense. It is also convenient way to get rid of all coins by topping up your travel card.
Downsides of using public transportation are the limited space for luggage, sometimes longer travel time and the amount of walking, waiting or transfers if there is not a perfect route available. This far these have not been a problem for me, though. If you are a person who enjoys conversations with strangers, taxi drivers can also be a good source of local knowledge.
Taxis are good thing to have, but if I have a choice, I will rather choose a public transportation option as I find it more comfortable for me. Public transportation saves money and environment, and gives the opportunity to experience local life.
Tip for big cities metro: print a colour version of the metro map on A3 size and keep it with you all the time. Works faster and better than any app!