While I’ve never been to India, I’ve travelled in other countries of Southeast Asia. These are countries where you see vibrant colours everywhere. If it were I, I would shoot a reversal film because of the gorgeous results you get with such vibrant scenes. My preference is Fujifilm Provia, but many people rave about Velvia. If you can, try to shoot a test roll before you go.
Unless you are planning on using a slow zoom lens, or doing a lot of shooting at night or something, I would go with ISO 100. Though with a faster film, you might have more flexibility. Results are still great with higher ISO films, and you can always use a ND filter if necessary. (Unfortunately, some years ago Fujifilm decided to discontinue my very favourite film, Provia 400X, so you won’t find ISO 400 reversal film anymore.)
If you’re just going to be away for 4 weeks, you don’t need to be overly concerned about keeping the film cold. Though if you have a fridge in your hotel room, it wouldn’t do any harm to keep it in there. Don’t subject the film to conditions where condensation could form on the film, e.g. allow the film to “acclimatise” when moving from cool to hot environments, and keep it in ziplock bags or something to protect from condensation forming. Obviously if you are going to be there during monsoon season, keep your film and camera dry. Take a look at How do I prepare frozen/refrigerated film for use?
Some people get really paranoid about airport x-ray scanning, but there’s no need. Just ensure that you keep your film in your hand luggage. Checked luggage does indeed get subjected to stronger x-ray scanning. There was a test performed by the French civil aviation authority in 2010 and it showed that ISO 400 film can withstand 12-24 passes through hand luggage x-ray scanning, and ISO 100 film can withstand more than 24 passes.
One of my favourite photographers is Steve McCurry, who has photographed extensively in India, including during monsoon season (see photo below!). It’s certainly worth checking him out for some inspiration. (By the way, I read years ago in a Kodak leaflet that Steve McCurry shot Kodak E100G film, so if you like his stuff, it could be worth giving the new Kodak Ektachrome a look.)
Addendum… I wanted to revisit this answer to add a couple of points.
I would never discourage anyone from shooting film, but just be aware of how important metering is. With film, you don’t have the instant feedback that you get with digital, and you can quite easily ruin your results through poor metering, especially with reversal film. Advanced SLRs from the late 90s or early 2000s will have good, reliable in-camera metering. If you are using something more manual, make sure you get the metering right.
While I recommended a reversal film at the top of this answer, my current film preference is Kodak Portra. It’s less saturated than a reversal film, but personally I think it produces gorgeous results, and if anyone were to shoot an India trip on Kodak Portra, I think it would also give wonderful results. A lot of this comes down to personal preference, and it’s worth trying out the options in advance and being familiar with your own preferences.