Local Transport
For long-distance travel by rail, see Trains. This page covers the rest: city transport, intercity options, and driving.
Ride-hailing apps: your default
Uber and Ola work in most major cities. All accept UPI payments and international cards. Rapido runs bike taxis and autos, which are cheaper and much faster through city traffic. In Bangalore, the Namma Yatri app is another option the locals use. You need to install Uber and Ola minimum. Prices are fixed at the time of booking.
Warnings! Drivers sometimes call to ask your destination. If they feel the ride isn't worth it, they ask you to cancel. You should refuse. If you cancel, you will usually incur a penalty. Insist that the driver cancels. During peak hours, it is sometimes difficult to get a ride. You might have to try for 15 to 30 minutes before securing a booking. So try simultaneously on all the apps you have: Ola, Uber, and Rapido. Book the first available ride and cancel the others immediately (cancelling right away is normally free; it's cancelling after a driver has waited that costs). Because of these uncertainties, allow sufficient time for your travels. In cities like Bangalore, for example, leave 4 hours before a domestic flight and 6 before an international flight.
Auto-rickshaws
The three-wheeled auto is everywhere and cheap. In big cities, booking through an app is now the norm; street-hailing is fading. In towns, try to insist on a meter plus an amount (20 to 50% extra) for rides. In select cities like Mumbai, autos and cabs actually run on meters if you hail them directly. In Bangalore and Delhi, use the app, no point trying to haggle.
Metro
Delhi, Mumbai, Bangalore, Chennai, Kolkata and Hyderabad all have metro systems now: clean, cheap and air-conditioned. Sometimes crowded. Metros are way more reliable and faster in all big cities as they are immune to road traffic. In metros, your bag goes through an X-ray scanner at every entry point. There are one or more coaches reserved for women. Buy tickets via QR in the metro's app, at the counter directly, or get a rechargeable card at the counter if you plan to use it a lot.
Domestic flights
IndiGo, Air India, Akasa, SpiceJet are the major domestic carriers. They connect all major cities and are cheap. Often a first class train ticket will be way more expensive than a flight. Use Google to look for the flight you want and book directly on the airline's website. You might save a few bucks on intermediary sites, but it's not worth the trouble if you need to change or cancel.
Intercity buses
State transport corporations and private operators both run intercity routes, ranging from basic non-AC seating to very comfortable AC sleeper coaches. You may have to take a bus if train tickets aren't available (this is common if you don't book sufficiently in advance). Bus tickets are available typically with a day's notice.
Getting out of the airport
At some major airports, you can book a cab at the company's counter before exiting the airport. These options tend to be more expensive than airport taxis, Ola, or Uber, but they are convenient. Outside the airport, well-marked signs direct you to Airport Taxis and app-based cabs. Airport taxis do not require an app. App-based rides have dedicated pickup zones; look for boards directing you there. The touts at arrivals will offer you cab options. Don't fall for them. Join the airport taxi line or book through the app. For your first entry into the country, use the airport taxi.
Self-driving: don't
Legally possible: you need an International Driving Permit plus your home licence together. Practically, don't. Even hardened Indian citizens often prefer to book a cab because of the chaotic nature of Indian traffic and driving conditions. Theoretically, we drive on the left, but you will often see a cab or a truck with its headlights on, coming towards you! Use trains (book way in advance) and flights for long distances, app-based cabs and autos in cities, and a hired car with driver for regional day trips. The car-with-driver is a standard mode of transport for single or multi-day trips. They are more expensive than the trains but still affordable and convenient. Hotels can arrange one, though booking online (just Google from A to B cab) will save you some money.