Flying
Flying into India is pretty much the same as any other country, except that you'll probably have to apply for an e-visa. You do this online ahead of time and pay a small fee. Make sure that you have printed out the document and bring it to your flight - the airline may or may not require that you have it with you, and you may or may not need it when you arrive, best to be safe! It's just a piece of paper, not stapled into your passport or anything like that.
From the Airport
Once you get off the plane and through passport control (which may take a while, be prepared - go to the bathroom ahead of time!) you'll need to get to your hotel. For many people, that will mean getting a taxi. This can be difficult and annoying anywhere in the world - no one loves to scam tourists more than taxi drivers! It'll also vary depending on the airport you arrived in. If possible, use Uber to save yourself haggling over price, or from a taxi driver who doesn't understand where you want to go. Some airports may have shuttles that will cheaply take you to a hotel, or your hotel may have a pickup service. Do your research ahead of time!
Getting Around Town
Getting around in any large Indian city is an exercise in traffic. This is the same in many, many other asian cities. You'll spend a long time sitting in traffic. Have your phone charged, and pay extra if you need to in order to have air conditioning! Tuk-tuks are cheap, but are not climate controlled - so you'll be exposed to the heat and exhaust fumes - and have much less safety equipment in case of an accident. Even worse is a moped or motorcycle!
Inside or outside of large cities, it may be worth your while to hire a driver for the day to get you around. Ask other tourists who they've used, or ask your hotel. It can be a little pricey but the convenience is nice, especially if the driver knows the area well and speaks your language. But they'll often try to sell you on various tourist attractions you may not be interested in - so again, research your destination ahead of time and know what you want to do.
On Foot
If you want to see "real" India, getting out and walking is worth it - but be careful! Traffic is unpredictable, sidewalks are often crowded (or nonexistent) and watch your step. Pavement can be uneven, and remember to watch out for animal droppings. Dress down, avoid conspicuous jewelry. Keep your phone secure in your pocket or hand. I usually walk with one hand in my pocket (on my wallet) and my phone gripped tightly in my hand - on the opposite side of passing traffic. Like anywhere, be wary at night or in poorly lit areas. Travel with other people. Avoid animals - you'll encounter lots of dogs, cats, cows, etc - stay back.