How can I cheaply transport lots of baggage when travelling domestically in India when coming in or going out on an international flight?

You're in luck - some airlines in India offer significantly reduced excess luggage rates to those traveling internationally on a separate ticket.

Flying on an airline with reduced excess luggage fees

This is probably the best option, as you get to check the luggage with you and fly to your final destination.

GoAir

The airline I'd choose is GoAir, as their excess luggage rate (each way) for checking an additional 30 kg (plus the normal allowance of 15 kg) is a mere 1,400 rupees, whereas they charge 11,400 rupees to non-connecting passengers, or 12,000 if paid at the airport. Keep in mind that you must prepay the bag fee in advance, and you must choose the right option to make sure you get the reduced "international" rate. If you show up to the airport with the extra luggage, you'll be charged the airport rate.

The counter agent will ask for your ticket proof that you are flying out of the country within 24 hours, or have flown into the country within the last 24 hours. (Don't try to slip a forged ticket by the agent, as they will contact your international carrier or check its website to see if your ticket is valid and in the same name.)

In the past, they used to offer excess baggage free of charge to passengers like you, but they no longer do; instead they offer this reduced rate.

IndiGo

IndiGo has a significantly larger network with more frequent flights than GoAir. As such, you may find a more convenient flight via IndiGo instead. IndiGo does also have reduced bag fees for international connecting passengers, however it's not as cheap, at 3,000 rupees (which is still way cheaper than 11,400-13,500 for non-connecting passengers).

Like GoAir, you will have to prepay this fee in advance, and you will be asked for proof of international travel within 24 hours.

Vistara

An alternative option to a budget airline, if you prefer to fly on a full-service carrier, is Vistara. They offer a complimentary allowance of two 23 kg bags to premium economy passengers only, if you show proof of international travel within the last 24 hours. After some searching, I've found that premium economy fares on Vistara tend to cost around 2,000-2,500 rupees more than standard economy fares. Effectively, this extra charge not only allows you to check more bags, but also gives you a better experience on the plane. This is also something to consider if you're not willing to ride a shuttle or pay for a cab to get to a different terminal, as in major hubs, Vistara operates from main terminals rather than separate budget airline terminals.

If you're only checking one 23 kg bag, you can also purchase a standard (non-Lite) economy fare, which you can check free of charge upon showing your international ticket.

It's worth noting, however, that you can only use these free allowances if you're traveling on a revenue (paid) ticket; if you're flying on an award ticket or an industry ticket, you won't be able to.

Air India

Air India does not have a reduced fee policy for international connections on separate tickets, but it does have a higher standard limit of 25 kg (as opposed to most carriers' 15 kg). As such, if you're traveling with only one bag weighing 23 kg, you'll be able to take it on Air India without paying a fee. However, if you're taking two bags, you'll have to pay a hefty fee.

In any case

At many major hub airports (e.g. Delhi and Mumbai), budget airlines operate from a separate terminal from international and non-budget carriers.

At Delhi airport, budget airlines operate from Terminals 1 and 2, with the main international terminal being Terminal 3. Terminal 2 is in walking distance of Terminal 3, but to catch a flight at Terminal 1 you'll need to take a complimentary shuttle bus.

At Mumbai airport, the main international terminal is Terminal 2, with the budget airline terminal being Terminal 1. There is no free shuttle between the two terminals, so you'll need to pay for a cab ride to get between terminals. (There used to be one, back when all domestic flights operated from Terminal 1; however, after the new Terminal 2 was built and most interline partner airlines moved their domestic operations there, this was discontinued.)

Be sure to search up the terminal layout of the airport you plan on connecting at in order to check if there is a terminal change, and budget at least 5 hours of time if there is a terminal change involved, and if so, if there is or isn't a complimentary inter-terminal shuttle.

Traveling by train

I wouldn't recommend this. First of all, you need to take a cab ride to the train station. Also, while you can carry up to 40 kg in sleeper class or AC 3-tier, or 50 kg in AC 2-tier, the size limit of 55 cm (21.5 in) for the largest dimension means that you won't be able to carry large suitcases with you. Additionally, if you're also carrying a carry-on suitcase, this counts towards that allowance as well. Indian Railways is starting to get a bit more strict on people exceeding the limits lately.

Since the large suitcases are too large to fit in a train cabin, you'll need to book space in the luggage car, and the Indian train system is known for theft. Add to that the fact that the Indian train system is notorious for delays, and the added travel time from traveling by train rather than flying. It's just better to fly, because it's much more convenient, requiring at most a terminal change and a small fee for extra baggage.