Nothing needs to be trimmed off. Read the FAQ here and through all the discussions about pruning. There is really no such thing as 'suckers'. They are lateral branches and they do produce fruit.

The claim that you must remove what are called suckers is old way-outdated wives tales spread by folks who don't understand how tomatoes grow.

Dave

Also, the things you are calling branches are leaf stems. You do not want to trim these off unless they are on the bottom of a mature plant. This is what the plant uses to collect energy from the sun. They are pretty good to keep. The more energy the plant can collect and use from the sun, the more energy it can put into making new plant parts, which if all else is good, could be more tomatoes.

CH