Can I sign in with one Apple ID for the App Store and a different ID for Apple Music? I basically want to sign in with a US Apple ID for the App Store and an Indian ID for Apple Music.
Is it possible to do that without losing all the music saved using Apple Music?
Please let me know
The "Mac App Store" accessed through "App Store.app" uses a totally seperate login system from iTunes.app, so you can use different AppleIDs for the two of them.

Within iTunes.app, I use multiple different AppleIDs to purchase iOS apps in different regions, which means I sometimes need to sign out of one AppleID and sign into a different one when switching regions and purchaseing or updating iOS apps. I have not encountered any issues when pushing these apps to my iOS devices or when running them on the iOS devices.

The Apple Music part is trickier.  According to http://www.idownloadblog.com/2015/10/04/apple-music-foreign-japan-account-in-usa/ you cannot easily have multiple regions using multiple iCloud Music Libraries, swithching AppleIDs for this purpose is limited.

As I understand things: if you want to use a different AppleID to buy some iOS apps, this should not be a problem - your current setup will not be impacted. If you want to use a different AppleID to sign into Apple Music then you will loose access to your current AppleID Apple Music stuff until you switch back.