San Marzano are usually indeterminates rather than a bush variety. They're the classic Italian plum-shaped tom that you get in good quality imported tinned toms. They're better for sauces than, say, salads because of the high ratio of flesh to juice.
Yes, nip off the little sucker growths from around the base of the plant. And it's a good idea to keep a gap of around at least a foot between the soil and the lowest foliage. Fungal spores can and will fall from the foliage to the soil and can be splashed back up again when watering. The gap helps guard against this.
SM, like a lot of plum varieties, are prone to Blossom End Rot. I've had it on SM plants planted immediately beside other varieties that haven't been affected. BER is associated with plant stress, but, for some reason, plum varieties are also genetically prone to it. No one knows why.
Less is better with both water and fertiliser with toms. Certainly don't feed them before the first fruit appears.
These are some of my toms in the ground after 12 weeks. I planted them with a handful of pelleted chook poo and they haven't had any fertiliser since. The temps here are in the low-30s and I water them once a week but very deeply. They're loaded with fruit.
As I say, for toms, less is better with food and water.