
The trifoliate branches need to be removed and never allowed to grow. From your photo (green on green), its hard to pick out what needs to be there and what should be removed. You may discover you are removing more tree than you are keeping because trifoliate root stock is very aggressive.
But even grafted trees need a root flare. Dig it out.
Citrus roots are very shallow. Planted in a lawn, your tree is going to have to compete for nutrients and water. If you are using weed and feed type products, you may be harming your tree with the herbicide while supplying too much nitrogen. That would keep it from blooming.
What is wrong with the tallest stem? I can't tell but the leaves don't look right and the tip is dead.
Your original question was proper pruning. Citrus don't need pruning per se but remove root stock suckers, dead branches, water sprouts and anything that crosses through the middle of the tree. You didn't say what kind of lime but type will determine eventual shape and size.