Will be touring around India for 6 months and
plan to be semi self-sufficient when it comes
to doing repairs etc.

What would be some essential tools and spare
parts that I would be required to carry for a
new Royal Enfield Bullet 350cc?

Different models have different requirements.
My Electra for instance needs both a 24mm and a 30mm flat ring spanner to remove the rear wheel and also to adjust the chain. You will also need 14mm and 18mm spanners.
I suggest you do a full service and see what tools you needed to accomplish this. Also remove and replace both wheels. Get a puncture repair outfit and a set of tyre levers you can handle. ( might be good idea to get this from agricultural merchants as tractor tubes can take -very big- patches. For spares, spare inner tube, engine filter and all the washers and O rings needed to change it. spare set of cables and all the tools needed to replace them. Spare chain and extra spare split link. Make sure you you have the tools to adjust primary and rear chains. Spare set of bulbs. insulating tape, self amalgamating tape, silicon grease spray for insulating electrical connections, assortment of jubilee clips, wire both electrical and fence. mole grips. And a pair of centre stand feet. If the bike has a rubber connector to join the carb to cylinder head carry one of those. After the bike has run happily for 2/300 miles store the spark plug as a known good spare ( wrap well) and fit a new one. Run that until it starts to deteriorate then repeat the process. Run you chains as a pair. after 2k miles switch chains and clean and regrease the spare one at your leisure. That way you will reduce the need to adjust the chain and extend your range.
My toolkit weight more than doubled when I went from BMW to Enfield. Not for reliability issues, but the Enfield uses a different size spanner for nearly every nut/bolt. You will need a mix of ring, open and socket spanners.