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The answer above provides a list of major telecom operators in India and most of the times you should get good connectivity with them in major cities/towns.
I lived in the southern part of India and travelled frequently to Goa and other cities along the coast and from my personal experience, I can say that Airtel provides very good internet along with solid connectivity in most of the urban/remote areas. You also get free national roaming. However, the downside is that it's a bit pricey as compared to other operators.
Reliance Jio is launched recently and it is undoubtedly the most affordable option by offering free voice calls, no roaming charges and you pay around 150 rupees per month for 42 GB 4G data. I find the connectivity quite good in the cities however it is unreliable at times in remote areas and also when you are travelling on a train.
If you talk about remote connectivity, I would say that BSNL offers the best connectivity but here the drawback is the internet speed which isn't the best.
Talking about the free-roaming, you should get it covered by most of the operators now. I suggest you buy multiple sims(most of the locals do that) because there would be times when one SIM gives you better speed than the other depending on the region you are in. You can go with Airtel & Jio, you will be spending around 300 rs a month (4 euros) which is still a pretty good deal for unlimited 4G data and voice calls.
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telecom operator :Airtel , Plan : Rupees 448 for 84 days , unlimited national roaming , unlimited nationwide calls/ SMS , free data roaming 1.4 GB Data/ day.
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India has a few mobile telephony companies (ordered by name):
All these companies have a pan-India presence so you should receive good enough connectivity almost everywhere. The only exception is if you plan to stay in remote areas (forests, tribal areas, higher himalyas etc.). In such areas only the state run BSNL provides connectivity. As Goa-Kochi corridor is very accessible, you can opt for any of the above.
Buying a SIM in India is easy and you should be able to use it for as long as you like. You can even switch your operator by retaining the same number (a long drawn process) or buying a new SIM altogether. We already have a question about buying SIM card in Hyderabad which can provide you with pointers even though it deals with a separate area.
EDIT:
You will be changing the states (Kochi is in Kerala, you will travel through Karnataka and finally end up in Goa). Indian SIMs are tethered to the "home" circle (The state in which they were bought). Using them in other states can incur national roaming charges. These days most of the companies provide unlimited national roaming free or as a part of the SIM plan. Be sure to check if data/calls/SMS (incoming and outgoing) are free when using national roaming before you buy a SIM or you may end up paying considerably more.