Math, Theory and Psychology are useful in online and live as well as tourney and cash games. I think they make up the foundation of the NLH world.
Id tell your friend:
memorize what wins and in what order
read the beginning [the first 4 chapters] of The Mental Game of Poker (part 1)
read this post:
http://forumserver.twoplustwo.com/32...-book-1399203/
Get a good math book. I'd recommend Poker Math That Matter by Gaines or The Math of Hold'em by Moshman and Zare. PMtM is the better book; but it's thin. IIRC tMoH shows more actual math and has more about tourney play; but, it's not as well written. (since IDK if your friend is interested in cash or tourneys like you play I listed both)
Next get a good theory book. The Theory of Poker is great but dense and can be boring if you're not serious about your game. It also involves examples from games you may not play like 7 card stud and Razz. I think it's definitely worth reading and definitely worth playing those games for free online to just understand the basic rules so that you can get the most out of that wonderful book; but, if you aren't that dedicated, skip it.
You can always get NLH Theory and Practice and Professional NLH vol.1. I haven't read J Little's new theory book so idk about it.
Then get the tMGoP part 2 and read the parts out of both that you need most.
the Psychology of Poker by schoonmaker is highly respected also.
Then I'd say it's time to get into strategy and hand reading. It shouldn't be too hard to find out which books are about which type of games they want to take to a more advanced level.
Good Luck