Hard/Classic should be pretty good, I found it a bit on the easy side save for a few missions but I've beat other FE titles on their hardest difficulties so this was to be expected. For someone new to FE but not to SRPGs, I'd say it'd be the best way to go. Some of the hardest chapters are earlier ones so don't be surprised if you find it tough at first, there's not a crazy difficulty spike, the difficulty is pretty consistent all the way through.
Also, no you don't miss any essential lore. Some weapon names are named after past characters, and there may be a few minor references here and there, but you don't really miss anything story wise for this one. DLC and Spotpass teams are something else as they directly pit you against characters from past games (albeit a bit tweaked).
Enjoy the game ! One of the best in a while for me, and play it with headphones when possible ! Sounds, production value, combat scenes and depth of content are far better than previous FE games in my opinion (if you do like it, you may want to try the GC and Wii ones after that, I give the edge in story and gameplay to those titles). Awakening might just end up my GOTY, will have to see still though as there's tons of great games coming. But FE:Radiant Dawn is one my all time favourites and I've spent over 120 hours so far on this game, which is way more than I spend on the average game nowadays.