Annamalai Swami was a disciple of Ramana Maharshi, who after realizing his Self did a really beautiful job distilling and spreading Maharshi's teaching. He also was able to add to them from the perspective of someone who didn't realize the Self at the age of 16, like Maharshi, but instead spent his whole life working for it. The core of the teaching is that everything is the Self. Consciousness alone exists and everything is a manifestation of this single reality. This reality is inherently peaceful, happy, loving -- which means that who I truly am at my core is happiness and peace. But because of illusions and attachments to the unreal forms -- our body, mind, and the "external world" -- we suffer. Deep down everybody is yearning to know their true Self, but is going the wrong way. "What you are seeking is who is doing the seeking".
So how to get back home, and find who is doing the seeking? This is where Annamalai Swami shines:
#1 Remember the Self
Don't "be" the Self, for that is too ambitious for an early practitioner. Just remember your true nature as often as you can. Make it a goal to do so not just in meditation, but in the midst of all activities. Doing so is the most worthy activity of this life.
#2 Exert a lot of effort and practice constantly
Unlike many of the neo-advaitists, he does not claim that this is something you can just realize in a flash of insight. Given the vasanas and the deeply rooted, outward habituation, it requires an enormous effort to go the other way and tame the mind. "You have forcibly to drag your wandering attention back to the Self each time it shows an interest in going anywhere else". It is like a battlefield, and you need to constantly replace the bad thoughts with better ones.
#3 You are not body. According to Annamalai Swami, all thoughts and separation rest on one, often subconscious thought, "I am the body". The jnani, however, looks at the body like we look at clothes. Without the bodily identification, all other separation falls away. The best way to fight incessant I am the body thoughts is the constant affirmation "I am the Self, I am the Self, I am the Self", which will lesson its power to affect you.
#4 you are not do-er. He didn't talk about this too much, but when he did he made to sure to underscore how pernicious this idea is. According to him, the idea of being the do-er is the thought poisoning your life. Because when you think you are the do-er you will start to think about all the other things you can be doing.