While existing answers are correct in essentials, they do not account for restrictions imposed on domestic travel in the wake of Covid-19 pandemic by India.

Here’s what you have to do to fly domestically in India these days (Sources – NDTV news, BS article, GoAir FAQs):

Before Flying

1. Download and install Arogya Setu app. The app must show your status as green. If you can’t download the app, you need to fill a self declaration form at the airport which may prove to be a hassle.
2. See what regulations are at force at your destination. A list is at Air India website. There are similar lists on each state’s ministry of health website and at various travel portals and other airlines websites’. Ensure you fulfill all requirements before travelling.
3. Pack your bags correctly. Only 1 piece of check-in bag (weight varies by airlines) and 1 cabin bag (NOT a trolley bag) is allowed.
4. Check-in online before your flight. This is required. Keep the boarding pass handy on your mobile or as a print out.
5. Print your bag tag and attach to your bag. If you can’t print the tag, write your name PNR and flight number on a paper and attach it securely to your bag.

Getting to the terminal

1. Ensure you are wearing a mask at the airport and maintain social distancing.
2. Show your Arogya setu app status to staff outside (there ought to be a line for this and they will probably have a thermal scanner to check your temperature).
3. Then move to the CISF counter at entry of terminal – show your ID (See this question about valid ID) & boarding pass to the security guy.
4. Enter the terminal and go to your airlines counter for bag drop. Show your PNR and drop your bag. Airlines will print the bag tag if you haven’t done so and they will also provide you a printed boarding pass if you don’t have one.

Security

1. Move to the security check area (There are signs everywhere to point you to this).
2. Put your cabin bag, wallet, phone, jacket etc. into a tray for scanning. You will also undergo a personal security check although in a contactless way (mostly a metal detector)
3. Collect your stuff & proceed to your boarding gate. The boarding gate is shown on your boarding pass and also on the many information screens you find in the terminal. Just look for your flight number.

At the gate

Wait till they announce the departure of your flight. This is also shown on the information screen right at the gate (Look for status – BOARDING). The staff will most probably board you in batches calling you by the seat numbers.

When it is your turn, collect your safety kit (face shield, mask and sanitizer) and self scan your boarding pass at the counter. Then move inside the aircraft (or you may be taken to a bus that takes you to the plane).

Inside the plane

Keep you bag in the overhead bin and sit on your alloted seat.

At the destination

1. Get down from the plane and follow the signs (and other passengers) to the baggage collection area.
2. Look at the information screen to know which belt your flight’s luggage will be delivered.
3. Wait at the belt and collect your baggage.
4. Proceed to the exit.
5. Depending on your destination, you may have to undergo quarantine procedures. See the list indicated to at the beginning of the answer to know more about those rules.

There are many youtube videos which show most of the above steps. You can search for those to familiarize yourself with the process of travel.

As these changes are temporary and the situation will, hopefully, normalize at some point in the future, here’s the "normal" procedure for taking a domestic flight in India

Before Flying

1. Ensure your have a copy of the ticket either electronic (on your mobile etc.) or paper. If you have done online check-in (this is optional), you should carry the boarding pass.
2. Check baggage rules of your ticket and pack accordingly.

At the terminal

1. Head for the entrance and show your ticket/boarding pass and ID to the security
2. Enter the terminal and go to your airlines counter. If you have checked-in you go to the bag drop counter and if you haven’t you go to the regular check-in counter. Counters are clearly marked on the overhead information screens or signs
3. After check-in is done and your bags deposited, move to the security

Security

1. Remove your wallet, jacket, belts, loose change, mobile etc. and put them into the tray along with your carry on luggage
2. You will be frisked by security.
3. Collect your stuff and move to your baording gate.

At the gate

1. To know your gate look at your boarding pass or (better) look at the information screens.
2. Wait at the gate for boarding to begin. Verify flight number at the gate information screen
3. At boarding time, staff will scan your boarding pass and allow you to board.
4. You will either enter the aircraft or take a bus to the plane

In the plane

1. Sit on your allotted seat and keep your carry-on luggage in overhead bin

At the destination

1. Get down from the plane and follow others/signs to the baggage claim area.
2. If you have baggage to claim, look at information screens and go to the indicated belt. Collect your baggage.
3. Move to the exit (Clearly marked)

Airports around the world work in the same basic fashion:

Check in -> (Immigration if international) -> Security -> Gate -> Airplane

Some airports have a second security check upon entering the building itself. There are airports with multiple terminals, generally one airline is only in one terminal but not always. Also, international flights may operate out of a separate terminal.

To find your check in location look first for your airline, then for a sign with your flight number or destination and class of service. It’s not unusual to not have separate counters for different flights. At this point you hand over any luggage you are checking, you receive a piece of paper that is your boarding pass. This paper has one important piece of new information–your gate number.

You then proceed through security–your bags will be x-rayed, in most airports you will walk through a metal detector, but other technologies are also in use. In general any garment which covers normal street wear must be removed and sent through the x-ray. Look at what the other passengers are doing, some airports require shoes and belts to go through the x-ray also.

You then go to the gate listed on your boarding pass. When you get there doublecheck that it’s correct (see what flight is shown on the signs at the gate), changes occasionally happen. There will be screens around listing flight numbers, cities, gates and time.

Once you’re on the plane pay attention to the safety announcement even though you will see most people are ignoring it–other than the location of the exit doors it’s basically the same for every flight, experienced travelers already know what they’re going to say.

If you search YouTube for things like "flying first time India," there are a number of videos people have made to explain the process for first-time travelers. Here’s the first one I found quickly that starts in Ahmedabad, so you can see that airport, though specific details may be a little different depending on which airline you’re flying.

https://www.youtube.com/watch?v=8NeWDtyAbS8

Airports have lots of uniformed staff who can help give you directions if you’re unsure where to go.

You will want to check the rules for your particular airline to know what kinds of ID they accept.