Removing suckers is removing tomatoes. Tomatoes form and grow on suckers. The only place to remove suckers is where the climate is low light such as in the UK. If I removed suckers down here in Texas I couldn't make a crop because of sunscald. But you wanted BIG tomatoes, not a lot of tomatoes. The main thing to growing big tomatoes is sunlight. The more the better. From your picture it looks like you have big shade trees and the house itself will also add shade. Fertilizing is a bit tricky. Too much nitrogen and you will grow a big plant but not many fruits. Too little and you don't get big tomatoes but you get quite a few smaller ones. Also, varieties are of big concern. Just because they sell the seed or plants in your area doesn't necessarily mean that the plants will give good results. If I wanted to grow BIG tomatoes I would remove some of the blooms not the suckers they grow on. Please update your profile so we will know your hardiness zone and your location.