There are likely regional variations, too.
In Sri Lanka, with rice, for example, the rice will be a sort of flattish mound on the plate. Curries are dished on to the rice. The person eating (right hand only is the custom) will take a bit of one curry and smush it into the rice, spreading the flavour around in the rice, then take some between the fingers and pop it into the mouth, using the thumb to push it in. Then he'll take some more curry, smoosh it into the rice, and repeat. Meat is torn from the bone with the right hand and not eaten like North Americans would at a barbecue with bone at the mouth itself.
Some people mix their curries together in the rice, others don't so much. Personal preference. Curd sambols or raita can either be eaten separately, with rice, or mixed in with another curry as personal preference dictates.
With naan or other flatbreads, rip a piece off with your fingers and use it to surround a bit of curry, then pop that in your mouth, rinse and repeat. Use another piece to sop up gravy. And so on.
Honestly, even here, no one cares if you do it differently than they do.
In restaurants, it's perfectly acceptable and pretty usual to eat with a spoon and fork. When dining in a local's home, they may give you a spoon and fork, but they're generally impressed if you try to eat with your hands the same way they do, even if you make a mess of things. They're happy at the attempts.