This varies enormously by country, so do a little targeted homework. Some cuisines are naturally kind to vegetarians, with deep traditions of vegetable, legume, and dairy dishes; others treat meat or fish as the default in nearly everything, including stocks, sauces, and seasonings, which is where hidden ingredients trip people up. Fish sauce, shrimp paste, meat broths, and lard are the classic invisible offenders, so learn to ask about them specifically rather than just saying vegetarian.
A few phrases in the local language are your best tool: learn to say and show in writing that you do not eat meat, fish, or eggs as applicable. Translation apps and printable dietary cards work well for allergies too. Big cities almost everywhere now have dedicated vegetarian and vegan restaurants findable through apps like HappyCow, while rural areas need more planning.
For Indian travellers specifically, Indian restaurants exist in most major cities worldwide as a reliable fallback, and religious sites of several faiths often have vegetarian eateries nearby.