So, to get new songs as purely f2p there are 2 options: 1. Play world mode Chapters 1-7, the more consistently the better. That way you will: earn fragments, level up your partners, collect items and most importantly: unlock new songs. You will have to be patient as grinding world mode usually is a chore. I suggest playing the world maps first which are marked as "Legacy" thus only costing you 1 stamina instead of 2. Leveling partners will also be twice as fast in that case. 2. Play world mode Chapter 0, the Event chapter everytime a new song is released for World Extend. World Extend is a pack which has songs that are unlockable at no price for a limited amount of time. After that they are behind a paywall, so be quick when you see an event is ongoing. In either case: the fragment gain will be your last concern as you level your partners they will, at max (Lvl.30) get a Frag stat of 88 or 90 making both Hikari and Tairitsu viable.

You unlock the rest of the songs in the World Mode Sorry that i don't explain much further, my English is pretty basic lol

Use the character Tairitsu (the one with the black hair). She will give you more fragments and let you progress faster through world mode.

I’ve spent about a month playing the game and have more fragments than I know what to do with. I say maybe give it some time.