All i know is that you should do the highest level you can do in advent battles.
As far as powering up the best way is indeed the disaster raids. If memory serves there are 3 weapons per raid, 2 will have hp increase (Defender) and 1 will have attack increase (Assault). And the eidos that come from the raids are some of the best SRs as far as overall usability. As always you should figure out 1 or 2 elements to focus on in the beginning. This is doubly so if you are new and not wanting to invest money into the game. Don't spend your materials on the weapons and eidos from the stores. You're better off using those for elixirs and seeds. The best way to gain power is to keep active, participate in events and do raids when not doing events.