You got the greeting right pretty much. You can say it the way you said it no problem.
The most correct ways to say are either as-salaamu 'alaykum OR salaamun 'alaykum. But don't sweat that.
I really am not a fan of nor do i recommend saudi style salafi islam. but it is growing in influence and popularity, mainly because of saudi money. Be sure to keep an open mind and try to base your religion in the Quran as much as you can. Everything else, you can do as long as it doesn't contradict the Quran as far as you know. When you learn that it does contradict the Quran, leave it alone. If it's not in the Quran, but doesn't exactly contradict the Quran (like the hijaab/niqaab), then this is your choice.
Just make sure you keep God and love for him in your heart and keep your attention on the Quran and know that it is the source of the guidance you follow. Not all hadeeths are bad, but just remember that the Quran is our deen. Everywhere you go and with everything you read, take what is good and what vibes with the Quran and leave alone what does not, respect your mind, but be humble and do not take stances on what you have no knowledge of. Do not fear not following something you don't understand. The Quran says in 17:36; "And do not follow what you have no knowledge of. Indeed, the hearing, the sight and the heart, each of these may be questioned (on this matter)."