To prune or not to prune, that is the question. Some people throw a cage on their tomato plants and just let nature take it's course. This means that their tomato plants will grow into a bush with a dozen separate branches.
I prefer to prune off the suckers and secondary branches, most of the time. If it is still early in the season, I will let a tomato plant split into two and sometimes three, if it gets ahead of me and I miss one, but I try to take off any suckers that grow.This is my Gordon Graham tomato plant. I have let it split into three stems only because it got ahead of me when I wasn't watching.
Pruning makes the tomatoes plants grow taller, so most of my tomato plants are staked instead of caged. I have tried caging them but they just grow over the top of the cages and fall over. I have seen tall homemade tomato cages that will do the job well, however. I still prefer to stake them and prune off the suckers. I find that this makes the tomatoes larger, with less per plant, and easier to see and harvest. These are my Portugal tomatoes, staked and producing wonderful, large tomatoes!
We were blessed with a pile of strong metal fence posts that I have used in the garden. I have hammered these into the ground and strung heavy coated wire between them. This is where I am growing the tomatoes this year. I just tie the plants to the wire as they grow up. Since I rotate the plants every year, I won't be growing tomatoes on this wire next year. I will probably grow cukes and pole beans on it. There is always something I grow that has to go vertical.
Suckers are little stems that grow in the leaf nodes. If left alone, they will split the plant into separate stalks, each growing tall, making a bush. About twice a week, I play in the tomatoes and nip off the suckers and tie up the stalks. It's an enjoyable activity and gives me a chance to keep a close eye on them.
I also prune the leaves on my tomato plants. I don't cut them all off, just a few. I trim off the ones that touch the ground. I think this might help to keep slugs off the plants. I also cut off any that interfere with the development and room needed by growing baby tomatoes and I prune leaves to open up the plant and let light and air circulation into the fruit. I do think it is important to leave a few big leaves on the plants to make food.
The 'San Marzano' tomatoes grow huge leaves that cover the entire plant. They have to be cut back some.