The difference between the Gra and the Duties of the Heart in terms of trust in God is to the Duties of the Heart one is allowed to do effort, but not to the Gra. That is to say we have the verse in Mishlei [Proverbs 3] "Trust in God with all your heart and do not depend on your own intellect." The Gra learns from the end of that verse not to do effort. And in the commentary of Mishlei there is an note from Menachem Mendel from Shkolov [a famous disciple of the Gra which adds an idea that he heard in the name of the Gra.]
The Duties of the Heart on the other hand says one should trust in God, but do effort. Just that one should trust in God concerning the results of his efforts.
But it can happen that if one accepts on himself the yoke of service of God, and removes from himself the pleasures of this world, then God might take away the need for him to do effort to reach his needs.
The Alter of Navardok brings this opinion of the Gra in the name of the Ramban. He got this from Israel Salanter who printed a magazine in Vilnius called Tevuna. In that magazine he wrote an article where he brings this opinion of "no effort" in the name of the Ramban.
So your end result is to the Gra you are not supposed to do effort, and to the Duties of the Heart you are allowed to but it is better not to.
To both of them it is best to sit and learn Torah.
Personally, I should admit I found it hard to learn Torah. And I also find it hard to learn. A lot depends on the kind of synagogue in your area. The insane religious world are obstacles towards learning Torah. Generally they are dens of the sitra achra [dark side]. Lithuanian yeshivas are clearly going to be places where if you are accepted you can learn Torah. But in general they tend to be picky about whom they let in, [and they should be picky.] So there is nothing simple about how to answer the question how to go about learning Torah. Learning at home I have also found almost impossible. So while I think learning is important, I have not found a good solution to the question of how to go about it.
The Duties of the Heart on the other hand says one should trust in God, but do effort. Just that one should trust in God concerning the results of his efforts.
But it can happen that if one accepts on himself the yoke of service of God, and removes from himself the pleasures of this world, then God might take away the need for him to do effort to reach his needs.
The Alter of Navardok brings this opinion of the Gra in the name of the Ramban. He got this from Israel Salanter who printed a magazine in Vilnius called Tevuna. In that magazine he wrote an article where he brings this opinion of "no effort" in the name of the Ramban.
So your end result is to the Gra you are not supposed to do effort, and to the Duties of the Heart you are allowed to but it is better not to.
To both of them it is best to sit and learn Torah.
Personally, I should admit I found it hard to learn Torah. And I also find it hard to learn. A lot depends on the kind of synagogue in your area. The insane religious world are obstacles towards learning Torah. Generally they are dens of the sitra achra [dark side]. Lithuanian yeshivas are clearly going to be places where if you are accepted you can learn Torah. But in general they tend to be picky about whom they let in, [and they should be picky.] So there is nothing simple about how to answer the question how to go about learning Torah. Learning at home I have also found almost impossible. So while I think learning is important, I have not found a good solution to the question of how to go about it.