I'd suggest you to fully hop on the NJPW train from January when their Wrestlemania equivalent- Wrestle Kingdom 8 comes live from the Tokyo Dome.
As for your questions:
1. It's puroresu- so wrestling is the most important. Their storylines and rivalries mostly revolve around the title belts which are treated with very high prestige and sometimes just a good old fashioned rivalry to see who is the better man. In ring action is what matters most.
2. Just who the hell are you?
Lol, nah Okada is the guy in my avi and sig. He's just 25 years old, a legend in the making, easily top 3 in the world atm and New Japan's future. Also, possesses the most beautiful finishing move in wrestling today- the Rainmaker.
3. Mostly I just follow the ppvs. A day or two after their broadcast, they are usually uploaded on Youtube.
4. Well, they have their annual G1 tournament. This year was the 23rd edition and won by Tetsuya Naito. Winner of the G1 these days gets a MITB style contract where he can face the IWGP Heavyweight Champion at WrestleKingdom in the Tokyo Dome. So, in January we're getting Naito vs defending champion Okada.
Nakamura and Tanahashi have been like the Orton and Cena of NJPW for many years. Haven't crossed paths in like 3 years but they will finally clash at WrestleKingdom for the IWGP Intercontinental Championship.
Bullet Club is a heel stable of the gaijins- Prince Devitt, Karl Anderson, Young Bucks, Bad Luck Fale, Doc Gallows and Tama Tonga. They demand respect and cause interference in their stable members' matches. Devitt will defend his Junior Heavyweight Title against Kota Ibushi at the Dome. Young Bucks are the Junior Tag champs, they will be in a match too.
These are the important storylines I can think of. We're most likely also getting Katsuyori Shibata vs Hirooki Goto, who have had a great series of matches this year.
I suggest watching WrestleKingdom 7, Invasion Attack, Dominion, G1 Climax 23 (days 2,4,8,9 like flagsabbath mentioned), King of Pro Wrestling from 2013 first so you get familiar with the wrestlers and pick a favorite