Legend of Soleanna is essentially the very same thing as the Gold Medalist achievement, only collecting every Silver Medal is not required to get it. In order for the achievements to pop, the following conditions must be met...
All S Ranks in every character's Action Stages (50,000 points in most stages)
All S Ranks in every character's Action Stages on Hard Mode (50,000 points in most stages)
All S Ranks in every character's Town Stages (30,000 points in most stages)
All S Ranks in every character's Boss Stages (30,000 points in most stages)
The key to scoring as many points as possible is to collect as many rings as possible and finish the level with all of them in your possession with a decent time. Taking down groups of enemies for extra points can also help, as long as there is little to no risk of losing your rings.
This being said, if you get hit or die in the middle of the stage, you might as well restart, as your score and rings are set back to zero upon respawning.
A good tip if your playing as Sonic is take advantage of the gems he acquires throughout his story, especially the red gem. If you keep RT held down through the whole stage, you can finish the stage with otherwise impossible times, seriously boosting your points. Other helpful gems for this would be the blue gem, (turbo, getting you through the stage faster) purple gem, (shrink, with an added ability (possibly a glitch?) of infinite jumping) and sky gem. (throws a gem and launches you to roughly the spot where it landed)
Shadow and Silver don't have any access to these gems, so it's considerably more difficult to get theirs. Some of the same strategies apply for them though, (rings, time, and enemies) and the scoring works the same.
Bosses also have the same scoring strategies, but with much less emphasis on rings and much more emphasis on time. Tear the boss down as quickly as possible while minimizing your hit intake and an S Rank should be acquired fairly easily.
Town Missions are a mixed bag, but most can be finished with an S Rank with a quick completion time.
If there are any more questions, send me a message or comment on the solution saying exactly what you need help on, and I will try my hardest to get back to you. Happy achievement hunting. =)