How do I get rid of the new growth, 'volunteer' palms that start at the base of my palm trees without hurting the main palm tree?
That depends on the species of palm, the type of volunteer and age of the tree and seedling/sprout. If you have a suckering palm such as a Date Palm you need to physically remove the suckers as they occur. If it's a single trunk like a Sabal and the seedling aren't already woody then Roundup should work. Just be sure the lower trunk is lignified (IE not green)so that the herbicide won't be absorbed by it as well!
I am not sure what type of palms they are. I had no idea how MANY types of palm trees there are! 1 of the palms has grape-like sized maroon growths circling under the green leaves. They drop off, and it looks like that is what is starting the volunteer growth at the base of this palm. I have tried round-up on the leaves themselves, but it doesn't seem to phase them. They already have a woody-base and just continue to grow, even as I try to cut the leaves to the ground. The leaves on this palm have barbs that will pop a soccer ball if it lands in the plant. The other palm has no barbs, and the blades come up from the middle of the plant, unfold and have thinner blades. I have not seen the grape like seeds on this one yet, but it's lower to the ground. It's just started this new growth at the bottom of the plant. It does not look like it's coming from the base of the original plant, it looks like a new palm starting just beside the base. Thank you so much for your advice.
It's most likely a Date Palm. I wouldn't spray any new growth at the base as it may kill the parent plant. Time for the Loppers (long handle pruner) and the Sharpshooter (shovel) to remove the unwanted new growth. Pictures of the other palm (Barbless?) would get you a better answer. HTH Chris